500 Integrals

7/23/2019 500 Integrals

http://slidepdf.com/reader/full/500-integrals 1/124

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7/23/2019 500 Integrals


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7/23/2019 500 Integrals


7/23/2019 500 Integrals


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7/23/2019 500 Integrals


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7/23/2019 500 Integrals


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7/23/2019 500 Integrals


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7/23/2019 500 Integrals


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7/23/2019 500 Integrals


• H wxvhyf yooz t oy π π oy 9

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7/23/2019 500 Integrals


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7/23/2019 500 Integrals


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9#>8 Zdf lonhykzdb =9

7/23/2019 500 Integrals


9#>9"9#>7 Zdf lonhykzdb hcj gobikchzkocw om lonhykzdbw hcj hlnfiyhkg mvcgzkocw =9

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:#988 8=7

7/23/2019 500 Integrals


:#989 8=4

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4#9: Zdf lonhykzdb kczfnyhl lk$ t+ 9804#9:0 988

4#9:8 Kczfnyhl yfpyfwfczhzkocw 989

4#96 Zdf pyoihiklkz{ kczfnyhl, zdf Myfwcfl Kczfnyhlw $ + +$+,$, tG tW tέ , zdf fyyoy

mvcgzkoc fym$ t+, hcj zdf gobplfbfczhy{ fyyoy mvcgzkoc fymg$ t+

98:

4#960 Jfmkckzkoc 986

4#969 Kczfnyhl yfpyfwfczhzkocw 98=

4#7 Fvlfy‛w Kczfnyhlw om zdf Mkywz hcj Wfgocj Ekcjw hcj Mvcgzkocw Nfcfyhzfj i{ Zdfb 998

4#78 Zdf nhbbh mvcgzkoc $Fvlfy‛w kczfnyhl om zdf wfgocj ekcj+1 Η$ q+ 9994#787 997

4#77 Mvcgzkochl yflhzkocw kc}ol}kcn zdf nhbbh mvcgzkoc 99:

4#778 9964#77: 979

4#776 974

4#77; Phyzkgvlhy ]hlvfw1 Moy c h chzvyhl cvbify 97;4#76 Zdf Kcgobplfzf Nhbbh Mvcgzkoc 967

4#760 Jfmkckzkoc 96:4#768 966

4#769 Wpfgkhl ghwfw 96=

7/23/2019 500 Integrals


4#767 Kczfnyhl yfpyfwfczhzkocw 9=8

4#76= Mvcgzkochl yflhzkocw 9=7

4#76; Yflhzkocwdkpw ~kzd ozdfy mvcgzkocw 9=6

4#7= Zdf Pwk mvcgzkoc $ + tτ 9>7

4#7=> Fvlfy‛w gocwzhcz1 Kczfnyhl yfpyfwfczhzkocw 9>:

Yfmfyfcgfw 9>=

KcjftKcjft om Moybvlhw

9>494:

LKWZ OM KLLVWZYHZKOCW

Mknvyf 8# Ftpocfczkhl Mvcgzkoc $Nyo~zd+ 9

Mknvyf 9# Ftpocfczkhl Mvcgzkoc $Jfgh{+ :

Mknvyf 7# Chzvyhl Lonhykzdb Mvcgzkoc :

Mknvyf :# Nhbbh Mvcgzkoc =Mknvyf 6# Fyyoy Mvcgzkoc 80

Mknvyf =# Kbhnkchy{ Fyyoy Mvcgzkoc 80Mknvyf ># Ftpocfczkhl"kczfnyhl Mvcgzkoc 89

Mknvyf 4# Lonhykzdb"kczfnyhl Mvcgzkoc 8:

Mknvyf ;# Fvlfy‛w Gocwzhcz 8=Mknvyf 80# Ghzhlhc‛w Gocwzhcz 8>

7/23/2019 500 Integrals


7/23/2019 500 Integrals


BHZDFBHZKGHL - NYHPDKGHL WVBBHY[

OM WFLFGZFJFLFBFCZHY[ HCJ WPFGKHL MVCGZKOCW

NHBBH MVCGZKOC

7/23/2019 500 Integrals


9

FTPOCFCZKHL MVCGZKOCW

tf

tf ∙

Mknvyf 8# Ftpocfczkhl Mvcgzkoc $Nyo~zd+ Mknvyf 9# Ftpocfczkhl Mvcgzkoc $Jfgh{+

Cozhzkoc1 $ + $ + t tf t 8lcftp ∙22

Jfmkckzkocw - Lh~w om Ftpocfczw1

$ +$ + t t t

t{ { t

{

t { t

{ t { t

ihhi

hh

h

hh

hhh

hh

2

2

2

2

≠2

∙

%

0,80

h t t

h t t

h t t

h t t

cc

cc

fh

fhfh

fh

lc

lc

lc

lc

∙∙

∙∙

2

2

2

2

$ +U R $ +

$ + $ +U R $ +U R U R

$ + $ +kbhnkchy{8

8

9

8

888

8

k

hhh

hhhh

xxx

x

p p px

p

pxx p px

p

2∙

∙2∙2∙

222

$ +$ + tcc tc

tcc tc

t

hh

h

ff t

ff t

f t

lclc

lclc

lc

2222

2

: : : : 4: : : : >= ftp#moyifLfz fh

Lkbkzw1

8lkb

8lkb

0

0

2

2

∙

↙

↙

t

t

t

t

f

f

0lkb

lkb

2

%∓2

∙

%∓↙

%∓↙

t

t

t

t

f

f

0lkblkb

lkb

22

%∓2

%∓↙

∙

%∓↙

%∓↙

t

h

th

t

t

h

t

t

f

t

t

f

t

f

0lkb

lkb0

2∙

∙2∙

∙∙

∓↙

∙∙

↙

t

ff

hi t

ff

itht

t

itht

t

t t

7/23/2019 500 Integrals


7

Jfyk}hzk}fw1

cfh ,, hyf hwwvbfj gocwzhczw ftgfpz ~dfyf cozfj

: : : : : : : : 4: : : : : : : : >=

lc

lclc

lc

8lc

yvlf gdhkc

jt

jvf

jt

jf

jt

jvhh

jt

jf

jt

jh

jt

j}vv

jt

jv}v

jt

vj}v

jt

jf

jt

jv

vv

vhvv

}}}v}}

2

22

%222 ∙

U Rgocwzhczhcoz

lclc

lc

8

lc

lc

lc

lc

lc

8lc

c

h t thjc

jf

jc

jh

h

h

jt

jf

jt

hj

hhjt

jf

jt

jh

hhctjt

jf

jt

jh

c th t t

t

h t t

th t t

tch t t

c

cc

c

cc

22

∙22

22

22

∙

∙

htht

htht

t t

t t

hfjt

jf

hfjt

jf

fjt

jf

fjt

jf

∙∙

∙∙

∙2

2

∙2

2

Kcjfmkckzf Kczfnyhlw1

h

fjtf

h

fjtf

fjtffjtf

htht

htht

t t t t

∙∙

∙∙

∙2∥ 2∥

∙2∥ 2∥

• Wff Wfgzkoc 9#79 moy nfcfyhlkqfj kcjfmkckzf ftpocfczkhl kczfnyhlw om moyb, ∥ ¸¸

,jtf tc

htbftpyfwwfj kc

zfybw om kcgobplfzf nhbbh mvcgzkocw# Wvbbhykqfj kc KCGOBPLFZF NHBBH MVCGZKOCW#

•

Zdfwf ftpocfczkhl kczfnyhlw hyf vwfj omzfc kc zdf f}hlvhzkoc om jfmkckzf kczfnyhlw om flfbfczhy{ hcj wobfwpfgkhl mvcgzkocw#

Yfmfyfcgfw

• Ifyw, Ghlgvlvw# Gd# =, pp# 7=> mm# - 7>6 mm 5 Gd# >, pp# :67 mm#5 Gd# 4, pp# 6:> mm#

• Ghyy, 8;>0

• J~kndz, Zhilf om Kczfnyhlw, 8;=8

• Wpkfnfl, Gdhpw# > - 90

• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw# Wfgzw# 0#9:6, 8#9 $“Zdf Ftpocfczkhl Mvcgzkoc‒+, 9#08 $“Zdf ihwkg

kczfnyhlw‒+

7/23/2019 500 Integrals


:

LONHYKZDBKG MVCGZKOCW

$Ihwf f+

Mknvyf 7# Chzvyhl Lonhykzdb Mvcgzkocw, tlc - ,lc

8

t-

t

8

Cozhzkoc1

t t lclon 2 $chzvyhl lonhykzdb, ihwf f+

$ +bb t t lonlon 2

Jfmkckzkoc1 0,lon8

32∥ t tz

jz t

t

tlc

8

t

8

t

7/23/2019 500 Integrals


6

Pyopfyzkfw1

$ +

$ +

⎟⎟⎟

⎠

⎓

⎜⎜⎜

⎒

⎕

2⎟⎟ ⎠

⎓⎜⎜⎒

⎕ 2∙

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ 22%

∙2

%2

h

ii

h

hi

ih

{

th

{

t {i th

t{h { t {i th

{ t {

t

{ t t{

lclclclc

lclclclc

lonlonlon

lon lon lon

t t

tc t c

lon8

lon

lonlon

∙2

2

htf

htf

tf

tf

ht

ht

t

t

∙2

2

∙2

2

∙

∙

lc

lc

lc

lc

Lkbkzw1

∙∓2

%∓2

%↙

%∓↙

t

t

t

t

lonlkb

lonlkb

0

0lon

lkb

lonlkb

2

%∓2

%∓↙

%∓↙

h t

h

t

t

t

t

t

∙∓2

2

%↙

%∓↙

t t

t t

f

t

f

t

lonlkb

0lon

lkb

0 0lon

lkb

lonlkb

02

%∓2

%↙

%∓↙

t

f

t

f

t

t

t

t

$ +

88

lc

8lc

0+8lc$

∙2⎟ ⎠

⎓⎜⎒

⎕

2

2

f

f

Jfyk}hzk}fw1

0,8lc

≠2 t tjt

tj

$ + $ +ith

ithbi

jt

ithj bb

%

%2

% ∙8lclc $ +yvlfgdhkc

8lc

jt

jv

vjt

vj 2


∥ 2 t t

jtlc

COZF1 wff Wfgzkoc 9#79 moy wpfgkhl lonhykzdbkg mvcgzkoc‚zdf jklonhykzdb,

$ +$ +

$ + 8,Lk

8lcLk

899

9

<2

∙∙2

∘

∥ ∓

2

te

t t

jt t

t t

e

e

Yfmfyfcgfw

• Ifyw, Ghlgvlvw# Gd# =, pp# 76> mm#5 Gd# >, pp# :6: mm#5 Gd# 4, pp# 6:> mm#

• Ghyy, 8;>0


• Wpkfnfl, Gdhpw# > - 90

• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw# Wfgzw# 8#6 $“Zdf lonhykzdb‒+, 9#08 $“Zdf ihwkg kczfnyhlw‒+

• ôlmyhb Mvcgzkocw ~fiwkzf, dzzp1//mvcgzkocw#~olmyhb#gob/NhbbhIfzhFym/

7/23/2019 500 Integrals


=

NHBBH MVCGZKOC

Mknvyf :# Nhbbh Mvcgzkoc, $ + tΗ

Jfmkckzkoc1

$ + ,0

8 jz fz qz q ∙

∓∙

∥ 2Η yfhl 03 q UMoybvlh 4#780#8R

$ + ,8

lc8

0

8

jz z

q

q

∥ ∙

⎟ ⎠

⎓⎜⎒

⎕ 2Η yfhl 03 q UMoybvlh 4#789#8R

Kczfnyhl, Pyojvgz, hcj Wfykfw Yfpyfwfczhzkocw1

$ + ∥ ∓

∙∙2Η

0

8 jz fz q t qz t t, 0, yfhl 3 t q UMoybvlh 4#789#9R

$ +$ +

0yfhl ,

8

lkb(

lkb

88

3

⎟ ⎠

⎓⎜⎒

⎕ %

2

%

2Η

∏∏2

∓↙

2

∓↙ t

e

t t

p

e t t

p p t

p

e

t

p p

e

t

p UHyzkc, Moybvlh 9#>R

$ +$ +

0 yfhl ,lkb

8

88

8

8

ftpftp

88 8

3%

2%

⎟ ⎠

⎓⎜⎒

⎕ %

2%

⎟ ⎠

⎓⎜⎒

⎕

∙2Η ∏∏ ∏

2

∓

2

∓

2∓↙

t te

e

t

c

e

t

e

t

e

t

e

t

t

t t

c

e

t

e e c

t

η UMoybvlh 4#799R

$ +

$ + t

cq

c

e q

c

e Η

Η⎟

⎠

⎓⎜

⎒

⎕ %Η∏

∙

2

8- -

8

0

UWfgzkoc 4#776R

$ + 0 yfhl ,8lclclon8

3⎢⎬⎥

⎫⎤⎮

⎦⎭

⎪⎩⎧

⎨∙⎟

⎠

⎓⎜⎒

⎕ %%%∙2Η ∘

∓

2

te

t

e

t t t t

e

η UHyzkc, Moybvlh 9#;R

Pyopfyzkfw1

$ + $ +(8∙2Η t t $ +$ +

,8

t

t t

∙Η∙2∙Η t coz h chzvyhl

cvb#

$ + %∓2Η%↙

t t 0lkb $ + %∓2Η

%∓↙ t

tlkb

$ + 0≠Η t $ + $ + t t t Η2%Η 8 $ + ∙∓2Η∙↙

t t 0lkb $ + t

tΗ

∙∓↙lkb kw kcjfzfybkchzf

7/23/2019 500 Integrals


>

Jfyk}hzk}fw1

$ +$ +

$ + $ + $ + t tzjz fz jz ff

t

jz fz jt

j t

jt

tj

z tz z t

z t

ΤΗ22

∉

∉

22Η‹2Η

∥ ∥

∥ ∓

∙∙∓

∙∙

∓∙∙

0

8

0

lc8

0

8

lc

$ +

$ +

$ + 4#7==#8+$Moybvlh 8

gocwzhczwFvlfy.

lc80

Τ2

∙2

2Η‹ ∥ ∓

∙

η

zjz f z

$ +$ + $ +∥ ∓

∙∙2Η0

8 lc jz z fz tcz tc • $ + $ +

$ +$ + t

t

t

jt

tj Τ2

Η

Η‹2

Ηlc

$jknhbbh oy Pwk mvcgzkoc+, Moybvlh 4#770

• $ + $ +U R

$ + $ +$ + t

t

jt

tj

t tjt

j

Η

Τ∙2

Η

Η∙2⎟⎟

⎠

⎓⎜⎜⎒

⎕

Η 9

88

Yflhzkoc zo Kcgobplfzf Nhbbh Mvcgzkocw, $ + $ + th th ,-, Ηη 1

$ + $ + $ +

: : : 7: : : 98:7:98:7:98:7:98

NHBBHKCGOBPLFZFNHBBHGOBPLFZF

VPPFY

8

LO^FY

0

8

0

8

, ,

jz fz jz fz jz fz

th thh

z

t

hz t

hz h ∙∓

∙∙∙∙∓

∙

∥ ∥ ∥ %2

Η%2Η η

UMoybvlh 4#76=#7R

• COZF1 Yflhzkoc kw hjjkzk}f5 f#n#, $ + $ + $ + ,, thh th Η∙Η2η

Zyknocobfzykg Mvcgzkocw1

• Ifyw, Gd# =#


• Wpkfnfl, Gd# 6#• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Wfgzw# 8#7"8":, 4#78"4#76 - Kcjft

COZF oc $ + tΗ 1 Fbkl Hyzkc‛w iykfm 8;=: iooe‚Zdf Nhbbh Mvcgzkoc‚pyo}kjfw h gobplfzf wzhzfbfcz moy

yfhl }hykhilfw om zdkw wpfgkhl zyhcwgfcjfczhl mvcgzkoc ghllfj zdf gobplfzf nhbbh mvcgzkoc, hcj jfyk}fw zdf

mvcjhbfczhl bhzdfbhzkghl pyopfyzkfw oyknkchzfj kc zdf glhwwkghl 84zd

- 8;zd

gfczvy{ ~oyew om Fvlfy, Nhvww,

Lfnfcjyf, Ykfbhcc, Wzkylkcn, ^fkfywzyhww, hcj ozdfyw# Wff Hyzkc‛w iooe moy ozdfy jfmkckzkocw, zdfoyfbw, hcj

jfyk}hzkocw om $ + tΗ coz nk}fc kc zdkw flfbfczhy{ iooe# ^dkzzhefy - ^hzwoc kw hlwo yfgobbfcjfj moy

jfyk}hzkocw#

Yfmfyfcgfw

• Hiyhbo~kzq - Wzfnvc, Gd# =#

• Hyzkc, 8;=:• Ifyw, Ghlgvlvw , Gdhpw# : - =, pp# :09"7

• Ghyy, 8;>0

• Boll, dzzp1//~~~#bhzd#zvlhcf#fjv/s}db/Zhilf#dzbl

• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Wfgzw 8#7"8#: hcj Wfgzw# 4#78"4#76 - Kcjft

• ^dkzzhefy - hzwoc, 8;7:

• ôlmyhb Mvcgzkocw ~fiwkzf, dzzp1//mvcgzkocw#~olmyhb#gob/NhbbhIfzhFym/

• O‛Iykfc, dzzp1//~~~#jogwzog#gob/jogw/6:>79>=/Pogddhbbfy"W{biol"Wflfgzfj"Pyoomw

• O‛Iykfc, dzzp1//~~~#jogwzog#gob/jogw/647=>47/Wflfgzfj"Zyhcwmoybhzkocw"Kjfczkzkfw""hcj"Wpfgkhl"

]hlvfw""moy"zdf"Nhbbh"Mvcgzkoc

7/23/2019 500 Integrals


4

KCGOBPLFZF NHBBH MVCGZKOCW

8# Lo~fy Kcgobplfzf Nhbbh Mvcgzkoc

Jfmkckzkoc1 $ + jz fz th z t

h ∙∙∥ 20

8,η , yfhl h 3 0 UMoybvlh 4#760#8R

COZF1

• Wff Wfgz# 4#769 moy kczfnfy wpfgkhl ghwfw, $ + tc,η , fzg#

Kczfnyhl Yfpyfwfczhzkoc1 $ + ∥ 32 ∙∙8

0

8 0 yfhl ,, h,tjz fz t th tz hhη UWfgzkocw 7#778 - 4#767R

Pyopfyzkfw1

• $ + $ + $ + ,, thh th Η∙Η2η

• $ +$ +

0,,8

, ≠%%

2∙

h

h

f t th th

thη η UMoybvlh 4#76=#8R

• $ + $ + t t έ2 π η 9

98 , , U $ + 2έ t fym$ t+, fyyoy mvcgzkoc, 4#960#8R

• $ + $ + t t έ2 π η ,98

• $ + 00, 2hη UMoybvlh 4#760#6R

• $ + $ +hh Η2∓,η

• $ + $ + thf t

jt

thj

jt

thj ∙∙2Η

∙2 8,,η UMoyb# 4#76=#:R

• $ + $ + $ +$ +yvlfgdhkc

,,,

jt

jv

jv

vhj

jt

vhj

jt

vhj η η 2

Η∙2

• $ + $ + ∥ ∙∙%2

8

0

8 lclc,, zjz fz t t thjh

thj tz hhη η UWfgz# 4#76=R

9# Vppfy Kcgobplfzf Nhbbh Mvcgzkoc

Jfmkckzkoc1 $ + jz fz thz

t

h ∙∓

∙∥ 2Η 8, UMoybvlh 4#760#9R

COZF1

• Wff Wfgz# 4#769 moy kczfnfy wpfgkhl ghwfw, $ + tc,Η , fzg#

Kczfnyhl Yfpyfwfczhzkoc1 $ + ∥

∓∙∙

32Η8

8

0 yfhl ,, h,tjz fz t thtz hh

UWfgzkocw 7#778 - 4#767R

Pyopfyzkfw1

• $ + $ + $ + ,, thh th η ∙Η2Η

• $ +$ +

,,8

,h

f t th th

th ∙∙%Η2Η 0≠h UMoybvlh 4#76=#9R

• $ + $ + t t έ∙2Η π π 9

98 , U $ + 2έ t fym$ t+, fyyoy mvcgzkoc, 4#960#8R

• $ + $ + t t έ∙2Η π π ,98

7/23/2019 500 Integrals


;

• $ + $ +hh Η2Η 0, UMoybvlh 4#760#7R

• $ + 0, 2∓Η h UMoybvlh 4#760#:R

• $ + $ +jt

vhj

jt

thj ,-

, ΗΗ$wff 8# hio}f+

• $ +$ + ∥

∓∙∙%Η2

Η

8

8 lclc,,

zjz fz t t thjh

thj tz hh UWfgz# 4#76=R

$ +$ +

$ +U R $ +$ + $ + ∥ 3

Η%

ΗΤ∙2⎦

⎭

⎪⎩⎧

⎨

Η∙∙

8

0

8 0yfhl ,lc,

lc,

h,tzjz fz h

t

h

thh t

h

th

jh

j tz hhη η

UWfgzkoc 4#76=R

COZF1

• Wff ftpocfczkhl"kczfnyhl mvcgzkoc, +$Fk t , moy yflhzkoc zo $ + th,Η


$ + $ + $ +

c

bjz fz

chch

htjtf t

z

ht

chtb

c

c 8

,

8,8 8

88%

2∙

2∙Η∙

2 ∙∓

∙

∙∙∙

∥ ∥ η

η η

η

η

η

η

$ +c

bjz fz

cc

tjtf t z

t

c tb

c

c 8,

8, 8 %2∙2

Η∙2 ∙

∓∙∙

∥ ∥ η ι ι

ι η

ι

η

η η

ι UMoybvlh 9#77#80R

$ + $ + $ +c

b qjz f

z c

h

c

ht qhjt

t

f z

ht

q

q qc q q

b

ht

c

c

8,

88,88

88 ∙2

∙2

∙∙Η∙2 ∙

∓

∙

%

%%

∥ ∥ UMoybvlh 9#796#=R

$ +c

b qjz

z

f

cc

t qjt

t

f

c

c

t

q

z qc q

b

t 8,

,8

∙2∙2

∙Η∙2 ∥ ∥

∓

%

∙∙

ι

ι ι ι ι UMoybvlh 9#77#8;R

COZF1

• Zdfwf flfbfczhy{ hlnfiyhkg"ftpocfczkhl moybvlhw hyf vwfj ftzfcwk}fl{ kc zdkw iooe zo jfyk}f zdf jfmkckzf kczfnyhlftpyfwwkocw $wff Moybvlhw‚7#79=#9, 7#:=9#8;, 7#748#4"7#748#80 - ozdfyw kc zdowf wfgzkocw‚moy cfnhzk}f

ftpocfczkhl moybw+, zo pyo}f ftkwzkcn moybvlhw, hcj zo gyfhzf cf~ kczfnyhlw moy h vwfmvl glhww om zyhcwgfcjfczhl

mvcgzkocw hcj wpfgkhl mvcgzkocw#

Wff Wfgzkoc 4#76; moy wpfgkhl moybw, ⎟ ⎠

⎓⎜⎒

⎕ ¸⎟

⎠

⎓⎜⎒

⎕ ¸¸Η t t ,

9

8-,

9

8η

Yfmfyfcgfw

• Hiyhbo~kzq - Wzfnvc, Gd# =

• Iohw, Gd# 88


• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Vwf om zdf Zhilfw $“Zdf mhgzoykhl nhbbh‒+, Wfgz# 4#76

• ôlmyhb Mvcgzkocw ~fiwkzf, dzzp1//mvcgzkocw#~olmyhb#gob/NhbbhIfzhFym/Nhbbh9/

7/23/2019 500 Integrals


80

PYOIHIKLKZ[ KCZFNYHL OY FYYOY MVCGZKOC $FYM+

HCJ

KBHNKCHY[ FYYOY MVCGZKOC $FYMK+

Mknvyf 6# Fyyoy Mvcgzkoc $fym+ Mknvyf =# Kbhnkchy{ Fyyoy Mvcgzkoc $fymk+

Jfmkckzkoc1

$ + $ + ∥ ∙22έ

t

z jz f t t0

99fym

π UMoybvlh 4#960#8R

$ + $ + ∥ ∓

∙2έ∙2 t

z jz f t t99

8fymgπ

UGobplkbfczhy{ Fyyoy Mvcgzkoc, Moybvlh 4#960#:R

Kczfnyhl Yfpyfwfczhzkoc1

$ + $ + ∥ ∙

22έ

9

0

8fym

t z

jz z

f t t

π UMoybvlh 4#968#8R

Pyopfyzkfw1

$ + $ + t t έ∙2∙έ

$ + 00 2έ

$ + 8¸2∓¸έ

t t

Wfgz# 4#76;

7/23/2019 500 Integrals


88

Yflhzkoc zo Kcgobplfzf Nhbbh Mvcgzkoc1

$ +π

η ⎟ ⎠

⎓⎜⎒

⎕

2έ

9,9

8 t

t $ +π

η ⎟ ⎠

⎓⎜⎒

⎕

2έ

t

t

,9

8

Kbhnkchy{ Fyyoy Mvcgzkoc, fymk$ q+1

$ +

∥ ∥ ∥ ∙

∙ 222

2

9

99

000

899+fymk$

+fym$+fymk$

kq z kq

z

q

z jz z

f

kjz f

kjz f q

k

kq q

π π π


$ +π

η

k

q

q

⎟ ⎠

⎓⎜⎒

⎕ ∙

2

9,9

8

fymk


$ +

$ +z jz f

z jz f

z

z

fymk9

fym 9

9

9

π

π

∥

∥

2

2∙

Yfmfyfcgfw

• Hiyhbo~kzq - Wzfnvc, Gd# >#

• Iohw, Gd# 88#



• Wpkfnfl, Gd# 76

• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Vwf om zdf Zhilfw $“Ftpocfczkhl hcj yflhzfj mvcgzkocw‒+, hcjWfgz# 4#960#

• ôlmyhb Mvcgzkocw ~fiwkzf, dzzp1//mvcgzkocw#~olmyhb#gob/NhbbhIfzhFym/Fym9/

7/23/2019 500 Integrals


89

FTPOCFCZKHL"KCZFNYHL MVCGZKOC

+$Fk t

Mknvyf ># Ftpocfczkhl"kczfnyhl Mvcgzkoc, Fk$t+, tf

t∙

∙

Jfmkckzkoc1

Moybvlhw 4#988#8 - 4#988#9

$ + 0 ,Fk <2∙2 ∥ ∥ ∓∙

∓

∙

∙

tjz z

fjz

z

f t

t z

t

z

$ + 0 ,lkbFk0

3⎦⎭

⎪⎩⎧

⎨%∙2 ∥ ∥

∓ ∙∙

∙

∙

%↙ tjz

z

fjz

z

f t

z

t

z

ζ

ζ

ζ $Ghvgd{ Pykcgkphl ]hlvf P]+

Pyopfyzkfw1

$ + %∓2∓%Fk $ + 0Fk 2∓∙

$ +0Fk kw coz jfmkcfj

Yflhzkoc zo Lonhykzdb"Kczfnyhl Mvcgzkoc1

$ + $ +

$ + $ + 0,0,lkFk

0,lkFk

≠<2

<2

h tfht

tf t

ht

t

t

7/23/2019 500 Integrals


87


$ + $ + tjz z

f t

t

z

∙Η∙2∙2 ∥ ∓

∙

∙

,0Fk $ + $ + tjz z

f t

t

z

,0Fk Η∙2∙2∙ ∥ ∓ ∙

$ + $ + tjz z

f

t t

z

∙Η22∙ ∥

∓

∙

∙

,0Fk $ + $ + tjz z

f

t t

z

,0Fk Η22∙∙ ∥

∓ ∙

Moyb# 4#76;#8


$ +wjww

fw

¸2∥ ¸

Fk

Yfmfyfcgfw

• Hiyhbo~kzq - Wzfnvc, Gd# 6#• Boll, dzzp1//~~~#bhzd#zvlhcf#fjv/s}db/Zhilf#dzbl

• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Vwf om zdf Zhilfw $“Ftpocfczkhl hcj yflhzfj mvcgzkocw‒+, hcj

Wfgz# 4#98 - Wfgzw# 7#0:"7#06 $“Kbpyopfy Kczfnyhlw‒5 “Pykcgkphl ]hlvfw‒+

• ^fkwwzfkc, Fykg ^# )Ftpocfczkhl Kczfnyhl#) Myob Bhzdôylj ""H ôlmyhb ^fi Yfwovygf#dzzp1//bhzd~oylj#~olmyhb#gob/FtpocfczkhlKczfnyhl#dzbl

• ôlmyhb Mvcgzkocw ~fiwkzf, dzzp1//mvcgzkocw#~olmyhb#gob/NhbbhIfzhFym/FtpKczfnyhlF/

7/23/2019 500 Integrals


8:

LONHYKZDB"KCZFNYHL MVCGZKOC

Mknvyf 4# Lonhykzdb"kczfnyhl Mvcgzkoc, lk$t+

Jfmkckzkoc1

Moybvlhw 4#9:0#8 - 4#9:0#9

$ + 8,lc

lk0

<2 ∥ tz

jz t

t

$ + $ + 8 ,lcFklclc

lkblk8

8

00

32⎦⎭

⎪⎩⎧

⎨%2 ∥ ∥

%

∙

↙ t t

z

jz

z

jz t

t

ζ

ζ

ζ $Ghvgd{ Pykcgkphl ]hlvf P]+

Pyopfyzkfw1

$ + 00lk 2 $ + %∓2∓%lk $ + jfmkcfjcoz8lk

Yflhzkoc zo Ftpocfczkhl"Kczfnyhl Mvcgzkoc1

$ + $ +

$ + $ + 0,8,lcFklk

8,lcFklk

≠<2

<2

h t th t

t t t

h


$ + $ + t t

t lc,08

lc,0lk ∙Η2⎟ ⎠

⎓⎜⎒

⎕ Η2∙ UMoybvlh 4#76;#9R

7/23/2019 500 Integrals


86


$ +ww

jwlk

lc2∥

Yfmfyfcgfw

• Hiyhbo~kzq - Wzfnvc, Gd# 6#


• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Vwf om zdf Zhilfw $“Ftpocfczkhl hcj yflhzfj mvcgzkocw‒+, Wfgzw#

4#9: - Wfgzw# 7#0:"7#06 $“Kbpyopfy Kczfnyhlw‒5 “Pykcgkphl ]hlvfw‒+

• ^fkwwzfkc, Fykg ^# )Lonhykzdbkg Kczfnyhl#) Myob Bhzdôylj ""H ôlmyhb ^fi Yfwovygf#dzzp1//bhzd~oylj#~olmyhb#gob/LonhykzdbkgKczfnyhl#dzbl

7/23/2019 500 Integrals


8=

FVLFY‛W GOCWZHCZ$ +η oyG

Mknvyf ;# Fvlfy‛w Gocwzhcz

Jfmkckzkoc1

⎟ ⎠

⎓⎜⎒

⎕ ∙%%%%2⎦

⎭

⎪⎩⎧

⎨∙2

∓↙

∙

2∓↙

∘ cc

ce c

c

e c

lc8

7

8

9

88lkblc

8lkb

8

8

Eη UMoybvlh 4#7=>#8R

Kczfnyhl Yfpyfwfczhzkocw1

jz z

jt t t ∥ ∥ ⎟ ⎠

⎓⎜⎒

⎕ ∙2∙∙2

∓ 8

00

8lclc+lc$+ftp$η UBhc{ ozdfy kczfnyhl yfpyfwfczhzkocwR

Pyopfyzkfw1

###==:9860#6>>oy 2η G

Yflhzkoc zo Gobplfzf Nhbbh Mvcgzkoc hcj Pwk $Jknhbbh+ Mvcgzkoc1

$ + $ + η ∙2Τ2Η‹ 88

Yfmfyfcgfw• Hiyhbo~kzq - Wzfnvc, Gd# 97#

• Ifyw, Ghlgvlvw, pp# 689"7


• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Vwf om zdf Zhilfw, hcj Wfgz# 4#7=> - Kcjft

• ^fkwwzfkc, Fykg ^# )Fvlfy"Bhwgdfyock Gocwzhcz#) Myob Bhzdôylj ""H ôlmyhb ^fi Yfwovygf#dzzp1//bhzd~oylj#~olmyhb#gob/Fvlfy"BhwgdfyockGocwzhcz#dzbl

t

+lc$ tft∙∙

7/23/2019 500 Integrals


8>

GHZHLHC‛W GOCWZHCZ$ +E N oy

Mknvyf 80# Ghzhlhc‛w Gocwzhcz

Jfmkckzkoc1

$ +$ +

E%∙%∙2%

∙2 ∘

∓

29999

09 >

8

6

8

7

8

8

8

89

8

b

b

bN UMoybvlh 0#97:#7R

Kczfnyhl Yfpyfwfczhzkocw1

$ +jt

t

tN ∥ 2

8

0

hygzhc UBhc{ ozdfy kczfnyhl yfpyfwfczhzkocwR

Pyopfyzkfw1

###6;:;=60#;86oy 2E N

Yfmfyfcgfw

• Hiyhbo~kzq - Wzfnvc, Gd# 97#


• Bhykgdf}, Olfn5 Wocjo~, Aochzdhc5 hcj ^fkwwzfkc, Fykg ^# )Ghzhlhc.w Gocwzhcz#) Myob Bhzdôylj""

H ôlmyhb ^fi Yfwovygf# dzzp1//bhzd~oylj#~olmyhb#gob/GhzhlhcwGocwzhcz#dzbl

• Wpkfnfl, Gd# 7:

• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Vwf om zdf Zhilfw, Moybvlh :#678#8, - Kcjft

t

$ + t

thygzhc

7/23/2019 500 Integrals


84

PHYZKHL MYHGZKOCW

Flfbfczhy{ Kjfczkzkfw

$ +

c

c

c

cc

h

th

t

h t th

t

t t

t

h t

t

h

h

th

h

t

z

tz t

t

z t

z

t t t

t

t

t t

t

z

z z

z

t

hh t

h

h t

t

t

h th

h

th

t

t

h th

h

th

t

z

tz

hz t

h

z

tz

h

z t

h

¸

¸

¸

¸¸

⎟ ⎠

⎓⎜⎒

⎕ %2

⎟ ⎠

⎓⎜⎒

⎕ %2¸

∙∙2

⎟ ⎠

⎓⎜⎒

⎕

⎟ ⎠

⎓⎜⎒

⎕

∙∙2

∙∙2

%

2%

∙2%

∙

2∙

∙

2

∙

∙

2∙

%2∙

%

2%

∙2%

∙2∙%2∙

∙

2∙∙

2∙

%

2%

∙2%

∙

2∙

%

2%

8

8

88

8

8

8

8

88

88

88

8

8

8

88

8

8

88

8

88

8

8

88

8

8

88

8

88

8

88

8

8

$ + $ +

$ + $ +

$ + $ +

$ +

$ +

$ +

$ +

⎟⎟⎟⎟

⎠

⎓

⎜⎜⎜⎜

⎒

⎕

%2

⎟⎟⎟⎟

⎠

⎓

⎜⎜⎜⎜

⎒

⎕

%2

%

∙2

∙

%

2%

∙∙

2∙

∙%2

∙

%∙2

%

%∙

2∙

∙∙

2∙

%∙2

%

∙∙2

∙

∙

∙

2

∙

%∙

2∙

∙∙

2∙

¸¸2

¸

¸¸2

¸

¸2

¸

ti

hi

ht

ihtiht

h

itith

h

vz

v

h

itith

h

iithi

h

ith

t

hiti

h

ihit

t

ithi

h

iith

t

z z vz vz

v

z vz vz z

v

t t t t

vz vz vz

vz vz vz z

z z z z

z z z z

h thhth t t

h th

z

ht

z

h t t

z

ithh

iz

ht

z

ith t

z

8

88

8

888

8

8

8

8

8

8

+$

+$

8

+$

8

88

88

8

88

+8$

8

88

+$

8

8

+$

8

8

8

8

+8$

8

8

8

8

+8$

8

888b

b

b

$ + $ +

$ + $ +

$ + $ + $ +

$ + $ + $ +

$ + $ +h t

t

h t

t

h t

t

ithiithi

h

ith

t

ithih

ithiith t

th

h

th th

t

th

z th

th th

z

z

vvz

v

vz

z

z

vz v

v

z v

z

z

vvz

v

vz

z

ith

i

ith t

h

t

ith

ith t

h

ith

i

t

ith

i

ith

ithi

h

ith

t

ithi

h

i

ith

ith

t

h t t

h

h th t t

h t t

h

h th t t

hz

hz

hz

%%

∙2

∙

∙∙

∙2

∙

%∙

%2

%

%∙

%2

%

%

∙%∙

%2

%

∙

2∙

%2∙

∙

2∙∙

2∙

%

2%

∙2%

∙∙

∙2

∙

%%%2

%

∙∙

∙2

∙

%∙

%2

%

∙∙

∙2

∙

%%%2%

∙

∙2

∙

99

8

8

99

9

99

99

99

99

9

99

9

9

99

9

9

99

9

8

8

8

88

8

88

8

88

88

88

8

7/23/2019 500 Integrals


8;

$ +

c

c

c

cc

h

th

t

h th t

¸

¸

¸

¸¸

⎟

⎠

⎓⎜

⎒

⎕ ∙2

⎟ ⎠

⎓⎜⎒

⎕ ∙2∙

8

8

⎟ ⎠

⎓⎜⎒

⎕ ¸2

⎟ ⎠

⎓⎜⎒

⎕ ¸2¸

h

ith

it

hitith

8

8

$ +$ +$ + $ +

$ +$ + $ + $ +

$ +$ +$ + $ +

i tihi

h tihh

i th t t

i tihi

h tihh

i th t t

i tihi

h tihh

i th t t

ihti

htihtiht

%%%∙%

2%∙

∙%%%%

2∙%

%∙∙%∙

2%%

∙2

∙2

∙ ⎟⎟

⎠

⎓

⎜⎜

⎒

⎕

⎟⎟

⎠

⎓

⎜⎜

⎒

⎕

88

88

88

8

88

8

888

$ + $ +

$ + $ +

$ + $ +

$ + $ +

$ + $ + thh thh th

h thh thh t

h th t

h th t

h t t

thh t

thh t

th t

th t

th t

th t

∙%%2∙

%∙

∙2

∙

%∙

∙2

∙

∙%

%2

∙

%∙∙2∙

98

988

98

988

99

99

99

99

99

99

99

99

9

COZF1 Zdfwf kjfczkzkfw omzfc wkbplkm{ kczfnyhl ghlgvlhzkocw

Wff Wfgzw# 9#79 - 7#7=7, iflo~, moy fthbplfw

Fthbplf1 NY Moybvlh 9#787#8, ∥ %

2bt

ifh

jt K #

Gdhcnf om }hykhilf1

bwjtjtbfjwfwbtbt 222 ,

$ +∥ %2

iwhw

jw

b K

8

Zdf myhgzkoc,$ +iwhw %

8, kw wkbplkmkfj i{ hio}f phyzkhl myhgzkoc,

$ + $ +ithh

iz

ht

z

ith t

z

%∙2

%,

~kzd 82z , wo zdhz$ +iwhw %

8⎟ ⎠

⎓⎜⎒

⎕

%∙2

iwh

i

wh

88#

Zdfc, #888

⎟ ⎠

⎓⎜⎒

⎕

%∙2⎟ ⎠

⎓⎜⎒

⎕

%∙2 ∥ ∥ ∥ jwiwh

i

w

jw

bhjwiwh

i

wbh K

Co~, hppl{ gdhcnf om }hykhilf, ,, ijwjz iwhz 2%2 ~dkgd lfhjw zo wolvzkoc,

$ + $ +U Rbt

btifhbt

bhz

w

bhz w

bhz

jz

w

jw

bhifh

jt K %∙2⎟

⎠

⎓⎜⎒

⎕ 2∙2⎟

⎠

⎓⎜⎒

⎕ ∙2

%2 ∥ ∥ ∥ lc

8lc

8lclc

88

Yfmfyfcgfw

• Ifyw, Ghlgvlvw, pp# :98 mm#

• Zhilf om Kczfnyhlw, Wfykfw, hcj Pyojvgzw, Wfgz# 9#8 $Yhzkochl Mvcgzkocw+#

7/23/2019 500 Integrals


90

BKWGFLLHCFOVW

Gobplfzkcn Zdf Wxvhyf

$ +

$ +

:

:

9

:

:

9

99

99

9

h

hgi

h

i th

h

hgi

h

i thgitht

∙∙⎟⎟

⎠

⎓⎜⎜⎒

⎕ %2

∙∙⎟ ⎠

⎓⎜⎒

⎕ %2%%

UZdf wfgocj moyb kw bowz vwfmvl moy kczfnyhlw#Wff Wfgzw# 9#79 - 7#797, iflo~, moy vwfwR

Mkckzf Ikcobkhl Ftphcwkocw

$ +

$ +

$ + $ + $ + $ +$ +

$ +

$ + $ + $ + ,(7

+9+$8$

(9

+8$

((

(88

(7

+9+$8$

(9

+8$

((

(

77998

00

77998

00

ccccc

c

a

a ac ac

a

a ac ac

ccccc

c

a

a acc

a

a acc

itithccc

ithcc

itchh

ith ac a

cith

a

cith

iihccc

ihcc

ichh

ih

ac a

cih

a

cih

%%∙∙

∙∙

%∙2

∙∙2∙⎟⎟

⎠

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%%∙∙

%∙

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∙

2⎟⎟

⎠

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⎒

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∙∙∙

2

∙

2

∙

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2

∙

2

∙

∘∘

∘∘

E

E

~dfyf ⎟⎟ ⎠

⎓⎜⎜⎒

⎕

a

ckw zdf ikcobkhl gofmmkgkfcz, jfmkcfj hw,

$ +$ + $ + a

acccc

ac a

c

a

c

L

L

798

898

+($(

(

⋆⋆%∙∙∙2

∙2⎟⎟

⎠ ⎓

⎜⎜⎒ ⎕ ,

moy mhgzoykhl ccccccc +8+$9$798897+9+$8$( ∙∙⋆⋆⋆2⋆⋆∙∙2 LL hcj, i{ jfmkckzkoc, 8(8(0 22

COZF1 Wzkylkcn hw{bpzozkg mhgzoykhl Moybvlh1 cfcc cc π 9s( ∙

Jovilf Mhgzoykhl

80,moy8ojj0,moy+9$678

f}fc0,moy+9$=:9

(( ⎡⎫

⎡

⎤

⎮

∙23∙⋆⋆

3∙⋆⋆

2cccc

ccc

c L

L

$ +

$ +$ + +9+$99$=:9((9+89+$79$678((89

+89+$89$678((89

cccccc

ccc

∙⋆⋆2 ∙∙⋆⋆2∙

%∙⋆⋆2%

LL

L

Cozf zdhz, i{ jfmkckzkoc, 8((8((0 2∙2

• Wff olmyhb fiwkzf, dzzp1//bhzd~oylj#~olmyhb#gob/JovilfMhgzoykhl#dzbl • O‛Iykfc, dzzp1//~~~#jogwzog#gob/jogw/6=0=89:/Jovilf"Mhgzoykhlw"Wflfgzfj"Pyoomw"hcj"Cozfw

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7/23/2019 500 Integrals


98

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Yfmfyfcgfw

• Ifyw, Ghlgvlvw, ]ol# 9, pp# 404 mm# - Iohw, Gd# :, pp# 977"97=

7/23/2019 500 Integrals


W[BIOL BFHCKCN

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99

7/23/2019 500 Integrals


KCZYOJVGZKOC

97

7/23/2019 500 Integrals


9:

7/23/2019 500 Integrals


‚Moybvlhw 0#888 - 0#889 - 0#898‚

Mkckzf Hykzdbfzkg Pyonyfwwkoc

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96

7/23/2019 500 Integrals


‚Moybvlhw 0#888 - 0#889 - 0#898‚

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7/23/2019 500 Integrals


KCJFMKCKZF KCZFNYHLW OM


9>

7/23/2019 500 Integrals


94

7/23/2019 500 Integrals


9;

7/23/2019 500 Integrals


‚‚Wfgz# 9#78‚‚



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η #

70

7/23/2019 500 Integrals


‚Wfgz#9#789‚



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U R#8,,lc 222 chbhh

78

7/23/2019 500 Integrals


‚Wfgz# 9#789‚

• 9cj kczfnyhl,$ +

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hjt

t

hi

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zo hio}f, hcj boyf nfcfyhl

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COZF1 Hw h gdfge oc ghlgvlhzkocw, km c 2 8, zdfc

79

7/23/2019 500 Integrals


‚Wfgz# 9#789‚

$ +$ + ∥ 222∙Η

∙∙2 #

lclc

8lc,8

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hf

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moy fyyoy mvcgzkoc, fym, ghlgvlhzkoc, jwfh

w

∥ ∙ 9

lc

8#

• }fykmkfj bhzdfbhzkgh

77

7/23/2019 500 Integrals


‚Wfgz# 9#789‚

COZF1 Hll wolvzkocw kc zdkw wfgzkoc ghc if }fykmkfj i{ jkmmfyfczkhzkcn vcjfy zdf kczfnyhl wknc# Zdyff

fthbplfw mollo~#

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f

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88

∙∙Η∙

2∙∥ #

Yfwzhzkcn $ + ,lc,8lc

∥ ∓

∙

∙∙2∙∙Ηh t

z hjz fz h th myob Bhzd# Wvbbhy{, hcj vwkcn Yvlf ' ; moy jkmmfyfczkhzkcn

kczfnyhlw UBhzd# Wvbbhy{R,$ +

$ +$ +

$ +$ + $ +

h

th th

h

h

h t

z h

h

h

t

hh t

jt

j fh t

hjz fz

hjt

j 2⎦⎭

⎪⎩⎧

⎨∙∙∙

∙2⎦

⎭

⎪⎩⎧

⎨ ∙ ∙

∙

∓

∙

∙∙∙ ∥ lclc

lc

8

lc

8 lc

8

lc

8

• 7yj ft# ‚]fykm{ wolvzkoc, $ + U R0lc lcfymklc9

89

≠2∥ h thh

jth t π

# Myob Bhzd Wvbbhy{

$Kcgobplfzf Nhbbh Mvcgzkocw+, yfwzhzf wolvzkoc hw,

$ + jz f

h

jz f

h

th

h

th

z

th

z

∥ ∥ 2⎟⎟

⎠

⎓

⎜⎜

⎒

⎕ 2

lc

0

lc

0

99

lc

89

lc9

8lcfymk

lc9

8

π

π π ,

wo zdhz, i{ Yvlf ' 4 moy jkmmfyfczkhzkcn kczfnyhlw UBhzd# Wvbbhy{R, hcj lh~w om ftpocfczw,

$ + 9999 lclc

lc

0

lc

lc

8

lc

8 th t th

th

z hf

jt

thj f

hjz f

hjt

j 22⎦

⎭

⎪⎩⎧

⎨2∥

‚‚‚‚Cozf1 Wff Wfgzkoc 4#76=, iflo~, moy jfyk}hzk}fw om hlnfiyhkg"ftpocfczkhl mvcgzkoc kczfnyhlw

Ftfygkwf1 ]fykm{ ozdfy wolvzkocw kc zdkw wfgzkoc#

7:

7/23/2019 500 Integrals


76

7/23/2019 500 Integrals


Wfgz# 9#79



U R0 lc

lc8

lc

8

≠⎦⎦⎭

⎪⎩⎩⎧

⎨¸

2

∥

∥ ¸¸

¸

∙

μ μ

μ μ

μ

jt t

tf p tf

tjtf

t t p

p t

p

b

• Kczfnyhzkoc i{ phyzw oc fhgd

kczfnyhl ~kzd, #,lc jtfj} tv t p μ ¸22

COZF1 wff jfm# kczfnyhlw iflo~ kc

Wfgzw# :#778, :#776#

Bhzdfbhzkgh govlj coz jo ghlgvlhzkoc‚

“Bhzdfbhzkgh govlj coz mkcj h moybvlh moy

{ovy kczfnyhl# Bowz lkefl{ zdkw bfhcw zdhz co

moybvlh ftkwzw#‒

$ +U R U R0 Fklc8

lc

≠¸¸

2

¸

¸

∥

μ μ μ

μ

μ

t tf

tjtf

t

t

b

• Wfz 82 p kc hio}f kczfnyhl

⎦⎭

⎪⎩⎧

⎨¸2 ∥ ∥

¸¸¸

jt t

tf p tf tjtf

t t p p t

μ μ μ

μ

8"plclc

8lc b

hcj ftpocfczkhl kczfnyhl, $ + t¸Fk , kc

Moyb# 9#77#84, Moyb# 9#796#>

o Oy‚kczfnyhzkoc i{ phyzw oc fhgd

kczfnyhl ~kzd, ,,lc jtfj} tv tμ ¸22 hcj

hppl{ jfmkckzkoc om ftpocfczkhl kczfnyhl,

$ + t¸Fk

}fykmkfj bhzdfbhzkgh#gob

Moybvlh 9#796#:

$ +hth

t

fh

t

hf

t

fjt

t

f hthththt

Fk===7

79

97:%∙∙

∙2∥

Jfyk}fj myob ftphcwkoc Moybvlh $9#79:#9+ kc N

- Y =f, ~dkgd wff iflo~

jtf tch

cbf

ch

t

jtf t

cc

c

htcbhtcb

htb

∥

∥ ∙

∙% ∙%∙

2

88

• Vwf Gdhcnf om ]hykhilf - Kczfnyhzkoci{ Phyzw#

∥

∥

∙

∙

2

%222

2

#8

nfz,zo,8

,,wfz

5Lfz8+

8

8

jwf wch

K

c

bcht

jt

jwht w

jtf t K

w

cc

htb c

η

η

η

9+ Jo kczfnyhzkoc i{ phyzw oc ∥ ∙ 8

jwf w wη hcj

yf"ftpyfww kc zfybw om t# Lfz

$ + 5,58, 98 w w f}jwfj}jw wjv wv 22∙22 ∙∙ η η η

$ +

$ +U R

$ + $ + $ +∥

∥

∥ ∥

∙∙∙

∙∙

∙∙

∙%∙2

∙∙2

∙∙2∙

η

η

η

η

η η

η

η η

η

η

ch

jtfchtht

c

cbf

ch

ht

jwf wf wch

K

jwf wf w}jvv}

c

chtcc

htc

w w

w w

898

98

98

8

88

Zdfc,

8

7=

7/23/2019 500 Integrals


Wfgz# 9#79

jtf tch

cbf

ch

t cc htcbhtcb

∥ ∙

∙% ∙%∙2

8

8

Km c 2 8, Moyb# 9#798#8 mollo~w hw wpfgkhl

ghwf#

Moybvlh 9#77#:

jtf tch

cbf

ch

t

jtf t

cc

c

htcbhtcb

htb

∥

∥ ¸∙¸

∙%

¸

∙%¸

2

88

b

Whbf hw hio}f moy ∥ ∙ jtf t

c

htb hcj wkncw

hwwkncfj#

COZF1 hlnfiyhkg"ftpocfczkhl moybvlhw om

zdf moyb $ +cb ht t ¸¸ ftp nfcfyhlkqf bhc{

locn"wzhcjkcn moybvlhw ~kzd 82c hcj wol}f

bhc{ flfbfczhy{ kcjfmkckzf hcj jfmkckzf

kczfnyhlw#

$ + $ +

$ +c

b

jz fz ch

ch

htjtf t

z

ht

chtb

c

c

88

,8

8

8

8

%

2

∙

2

∙Η∙2

∙∓

∙

∙∙

∙

∥

∥

η

η

η

η

η

η

η

• Wff Wfgz# 9#78 moy jfyk}hzkoc5 zdfc hppl{

jfmkckzkoc om kcgobplfzf nhbbh mvcgzkoc#

• $ + ,••Η kw kcgobplfzf nhbbh mvcgzkoc,

$ + t,hwnk}fc4#760#9, Moybvlh θ Η Moybvlh obkzzfj kc NY>5 wfcz kc hw

fyyhzvb

]fykm{ jfyk}fj kczfnyhl i{ jkmmfyfczkhzkcn

zdf kczfnyhl vwkcn Yvlf ' 6 kc Bhzd#

Wvbbhy{ ~kzd $ + cht t ∙2τ 1

$ +$ +

$ +

$ + $ + $ + $ + cchtb

chtc

t

f tjt

htj fht

ch

jz z m jt

j

ch

2⎦

⎭

⎪⎩

⎧

⎨ ∙∙∙

∙2

∙

∙∙∙∙

∓∙

∥

88

8

8

,8

η

η

η

τ

η

η

η

}fykmkfj bhzdfbhzkgh#gob

Moybvlh 9#77#6

$ +$ +

$ +$ +

⎦⎭

⎪⎩⎧

⎨≠≠2

%2

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙

2

%

∙∙∙

2∘

∥

0,08,9,###,8

(8

(88

8

88

0

hcc

b

h

t

e c

f

jtf t

e

e c

e

e ht

htb

c

c

η

η

η η η

• Kbpyo}fj Moybvlh 9#77#6 moy kczfnfy η #

• #:4#769 Moyb# myob Jfyk}fj ,

$ + $ + ∘∙

2

∙ 2∙2Η8

0

,###R9,8U (

(8,c

b

b t c

b

tfc tc

• Yfwzhzf kczfnyhl hw kcgobplfzf nhbbh

mvcgzkoc, hw pyf}kovwl{ jkwglowfj $obkzzfj kc

NY >+1

$ + $ +c

b

ch

ht

jtf t

chtb c 8

,

,88 %

2∙Η∙

2

∥

∙

η

η

η

η

• Plvn kczo Moyb# 4#769#:1

$ + $ +

c

c

b

b t

ht tc

bc

cb

tfc tc

∙22%

2

2∙2Η ∘∙

2

∙

,8

kwjwviwzkzvzf~dfyf

,###R9,8U (

(8,8

0

η

zo nk}f1

7>

7/23/2019 500 Integrals


Wfgz# 9#79

$ + $ +$ +

∥ ∘

∙

2∙ ∙

∙∙2

(

(88

8

0

8

e

ht

ch

fjtf t e

e c

hthtb

c

c

η

η

η η

• Co~ ~ykzf wvbbhzkoc kc yf}fywf oyjfy hcj

wkbplkm{, cozkcn zdhz kzw mkywz zfyb kw powkzk}f,

$ + $ + $ +088 888 ∙2∙∙ ∙∙ η η 1

$ +

$ +$ +

$ +$ +

$ +$ +

$ +

$ + $ +$ + $ +$ +$ +

$ + $ +

⎦

⎦⎦⎦

⎭

⎪

⎩

⎩⎩⎩

⎧

⎨

∙∙

∙∙

∙∙%

∙∙

2

⎦⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩⎩

⎧

⎨

∙∙∙∙

%

∙∙

∙∙2

∙

∙∙∙

∙∙

∙∙

∥

η

η

η η η

η

η

η η

η

η

η η η

η

η η η

h

h

t

h

t

h

t

c

f

ht

htht

fch

jtf t

ccc

ht

c

cc

hthtb

c

cc

8

7

7

9

98

8

7

98

8(8

988

8(7

(9(8(8

E

E

$ +$ +

$ +$ +

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙2

%

∙∙∙

2∘ 8

88

0 (8

(88

e

e c

e

e ht

h

t

e c

fc η η

η

η

i{ kcwpfgzkoc, cozkcn yflhzkoc,

$ +$ +

$ +$ +$ + $ +

#

,798(8

(8

0moyvcjfywzooj kw8om }hvlfzdf~dfyf

0

2

∙∙∙∙2∙∙

∙ 3

e

e e

e

η η η η η

η E

COZF1 cf~ }fywkoc zyhcwlhzfw boyf yfhjkl{

kczo ghlgvlhzkocw moy ,:,7,9,82η nk}fc kc

Moybw# 9#77#= ‗ 9#77#;#

COZF1 Moybvlh kw vwfmvl moy ~ykzkcn zdf lon"

ftphcwkocw kc Wfgzkoc 9#> ~dkgd ghc if

zyhcwmoybfj myob lon# mvcgzkoc zo ftp#

mvcgzkoc om zdf moyb #chtbf t Zdyff $7+

fthbplfw myob Wfgz# 9#>1

8+ Moyb# 9#>88, zyhcwmoybfj zo

ftpocfczkhl moyb,

∥ ∥ 2 jwf w tjt wbblc i{ gdhcnf om

}hykhilf

,,,lc ⎟ ⎠

⎓⎜⎒

⎕ 222 tf

t

jtjw t w w

~dfyf

b kw kczfnfy

• Hppl{kcn Moyb# 9#77#6,

74

7/23/2019 500 Integrals


Wfgz# 9#79

$ +$ +

$ +$ +

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙

%

∙∙∙

2∘ 8

88

0 (8

(88

e

e c

e

e ht

h

t

e c

fc η η

η

η

~kzd

,8,8,,8,lc 2%2↙222 hbbcc t t w η

zo nk}f,

$ +$ +

$ +

$ +$ +∘

∥ ∘∥

2

∙

2

∙2

∙∙2

∙∙22

b

e

e be

b

e

e be w wb t wb

te b

b t

we b

bfjwf w tjt

0

0

lc

,lc(

(8

# (

(8lc

~dkgd jkmmfyw myob Moyb# 9#>88 ocl{ kc

ftpyfwwkoc om mhgzoykhl zfyb#

9+ Moyb# 9#>99,$ +

∥ ∥ %2 jwf w tjt t wcbbc 8lc , i{

gdhcnf om }hykhilf

$ +⎟ ⎠ ⎓⎜

⎒ ⎕ 2222 %% wcc w f t tf

t

jtjw t w 88,,,lc

• Hppl{kcn

$ +$ +

$ +$ +

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙

%

∙∙∙

2∘ 8

88

0 (8

(88

e

e c

e

e ht

h

t

e c

fc η η

η

η

~kzd

,8,8,,8,lc %2%2↙222 chbbcc t t w η

zo nk}f,

$ +

$ + $ +$ +

$ +$ +

$ +$ + $ +∘

∘

∥ ∥

2%

∙%

2%

∙%

%

2

%∙∙2

%∙∙2

22

b

e e

e be c

b

e e

e be wc

wcb

t w

bc

c

t

e b

b t

c

w

e b

bf

jwf w tjt t

08

8

08

8

8

lc

,8

lc

(

(8

8(

(8

lc

~dkgd jkmmfyw myob Moyb# 9#>99 ocl{ kc

ftpyfwwkoc om mhgzoykhl zfyb#

7+ Moybvlh 9#>97#8,

$ +

∥ ∥

%2

2 jw wf tjt twc

t wc 8

lc

lc i{

gdhcnf om }hykhilf,

$ +⎟

⎠

⎓⎜⎒

⎕ 2222 %% wcc w f t tf

t

jtjw t w

88,,,lc

• Hppl{kcn

$ +$ +

$ +$ +

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙

%

∙∙∙

2∘ 8

88

0 (8

(88

e

e c

e

e ht

h

t

e c

fc η η

η

η

7;

7/23/2019 500 Integrals


Wfgz# 9#79

~kzd

,8,9,8,8,lc %222↙222 chbcc t t w η

zo nk}f,

$ +

$ +

$ +

$ +#

8

8

8

lc

8

8

8

lc

9

8

9

8

8lc

⎦⎭

⎪⎩⎧

⎨

%∙

%2

⎦⎭

⎪

⎩⎧

⎨

%∙%2

2

%

%

%2

∥ ∥

cc

t t

cc

w

f

jw wf tjt t

c

wc

wc t w

c

COZF1 Moyb# 9#798#9, ∥ ,jtf t htckw h

wpfgkhl ghwf om ∥ ,jtf tc

htb ~kzd

8,8, %222 cccb η #

Moybvlh 9#77#=

⎦⎭

⎪

⎩⎧

⎨2

%22

∥ 8

8

c

b

ch

htfjtf t

c

htb c

η

Moybvlh 9#77#>

⎦⎭

⎪⎩⎧

⎨2

%2⎟⎟

⎠

⎓⎜⎜⎒

⎕ ∙2∥ 9

889 c

bη

hh

t

c

htfjtf t

cc

htb c

Wviwzkzvzf }hlvfw om kczfnfy η kc 9#77#6

:0

7/23/2019 500 Integrals


Wfgz# 9#79

Moybvlh 9#77#4

⎦⎭

⎪⎩⎧

⎨2

%2

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ %∙2∥

78

9979

9

c

bη

hh

t

h

t

c

htf jtf t

ccc

htb c

Moybvlh 9#77#;

⎦⎭

⎪⎩⎧

⎨2

%2

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ ∙%∙2∥

:8

==7

:79

97

c

b

hh

t

h

t

h

t

c

f jtf t

ccchthtb

c

c

η

Moybvlh 9#77#9

U R th

jtfht hfymk9

89 π 2∥

• Vwf gdhcnf om }hykhilf, thz 2 - kcjfmkckzf kczfnyhl

jfmkckzkoc om $ +z fymk 1 #9

+$fymk9

∥ 2 jz fz z

π

Fymk jfyk}fj myob ∥ jtf tchtb

~kzd9

82η hcj Moybvlh

4#76;#:, ⎟ ⎠ ⎓⎜

⎒ ⎕ 9,

98 tη ~kzd #99 q t ∙↙

Moybvlh 9#796#87$ +

U R0

hfymkftp9

8 9

99

≠

⎟ ⎠

⎓⎜⎒

⎕ %⎟⎟

⎠

⎓⎜⎜⎒

⎕ ∙

2∥ %%

h

h

i t

h

ihg

h

jtf githt

π

Gobplfzf zdf wxvhyf oc ftpocfcz, vwf gdhcnf om }hykhilf,

hcj jfmkckzkoc om kbhnkchy{ fyyoy mvcgzkoc, fymk, myob hio}f#

• Zdvw, gobplfzkcn zdf wxvhyf hcj gdhcnkcn zdf }hykhilf1

∥

∥

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ ∙

2⎟⎟ ⎠

⎓⎜⎜⎒

⎕ %⎟⎟ ⎠

⎓⎜⎜⎒

⎕ ∙2

2%2∙

∙⎟⎟ ⎠

⎓⎜⎜⎒

⎕ %

2∙

∙⎟ ⎠

⎓⎜⎒

⎕ %2%%

jwfh

ihg

h

jth

i th

h

ihg K

hjt

jw

h

i th w

h

hgi

h

i th

h

hgi

h

i thgitht

w99

99

99

99

9

ftp8

ftpftp

555

:

::9

~dkgd lfhjw zo yfwvlz#

Bhzdfbhzkgh wolvzkoc hz dzzp1//kczfnyhlw#~olmyhb#gob/

wdo~w zdf h~fwobf wpffj om gobpvzfy pyogfwwkcn, hcj zdf

myfxvfcz cffj moy powz"pyogfwwkcn wkbplkmkghzkoc1

• GDFGE1 Km 022 gi kc$ +jtf githt

∥ %% 99

, zdf kczfnyhl

:8

7/23/2019 500 Integrals


Wfgz# 9#79

kw kjfczkghl zo U R th

jtfht hfymk9

89 π ∥ 2 om Moyb#

9#77#9#

• COZF1 wff Moyb# 9#77#8, NY >, moy h wkbklhy cfnhzk}f

ftpocfczkhl kczfnyhl5 vwfw whbf hppyohgd

kc}ol}kcn $ + #

9

fym

9

∥ ∙

2 jz f t

z

π

• COZF1 Wfgz# 7#797, iflo~, dhw bhc{ wkbklhy jfmkckzf

kczfnyhlw

• ]fykm{ wolvzkoc i{ jkmmfyfczkhzkcn zdf kczfnyhl

Moybvlh 9#77#80

$ +

c

bjz fz

c

c

tjtf t

z

t

c tb

c

c

88

,

8 %2∙2

Η∙2

∙∓

∙

∙

∥

∥

η ι

ι

ι η

ι

η

η

η

ι

• Wff Wfgz# 9#78 moy jfyk}hzkoc5 zdfc hppl{ jfmkckzkoc hcj

pyopfyzkfw om kcgobplfzf nhbbh mvcgzkoc

• $ + mvcgzkoc,nhbbhkcgobplfzfkw,••Η

$ + t,hwnk}fc,4;0p#9,4#760 Moybvlh θ Η

• COZF1 zdkw flfbfczhy{ kczfnyhl wol}fw hcj gyfhzfw

bhc{ kczfnyhlw kc Wfgzw# 7#7"7#:, iflo~#

• Ftfygkwfw1o ]fykm{ wolvzkoc i{ jkmmfyfczkhzkcn vcjfy zdf kczfnyhl

wknc

o Wdo~ zdhz zdf wolvzkoc,

$ +,,8 thjtf t th Η∙2∥ ∙∙

yfpyojvgfw zdf jfmkckzkocw kc

Bhzd# Wvbbhy{ om $ + $ + $ +5,,,, th thh η ΗΗ f#n#, wdo~

zdhz,

$ + $ +h thjtf t t

t th Η2Η∙2∥

∓

2

∓2∙∙

00

8 , i{ kcjfmkckzf kczfnyhl

wolvzkoc hcj pyopfyzkfw om nhbbh mvcgzkocw

• }fykmkfj bhzdfbhzkgh#gobMoybvlh 9#77#88

$ +

$ + $ +$ +

$ +

⎦⎭

⎪⎩⎧

⎨≠≠2

%2

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙∙

2∙

∘

∥ ∙

2%

∙∙

0,08,9,###,8

(8

(8ftp

ftp

8

08

8

ι η

ι η

η ι

ι

η η

cc

b

t

e c

t

jt t t

e e

e cc

cb

• Kbpyo}fj Moybvlh 9#77#88 moy kczfnfy #η

Wkbklhy kc jfyk}hzkoc zo hio}f kczfnfy"ftphcwkoc,

$ +$ +

$ +$ +

⎦⎭

⎪⎩⎧

⎨2

%2

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙2∥ ∘ %

∙∙∙

2

8,9,###8

(8

(88

8

88

0

c

b

h

t

e c

fjtf t

e

e c

e

e ht

htb

c

c

η

η

η η η

• Ihwfj oc kcgobplfzf nhbbh $wpfgkhl ghwf+ ftphcwkoc,

Moyb# 4#769#:,

$ + $ + ∘∙

2

∙ 2∙2Η

8

0

,###R9,8U (

(8,c

b

b

t cb tfc tc

~dfyf wviwzkzvzfj kw c t tc

bc ι η 22

%2 ,

8

• $h+ Wzhzf kczfnyhl hw kcgobplfzf nhbbh mvcgzkoc, Moyb

9#77#80, $i+ zyhcwlhzf kczo kcgobplfzf nhbbh $wpfgkhl

ghwf+, Moybvlh 4#769 :, cozkcn zdhz hll zfybw kc

wvbbhzkoc ftpyfwwkoc hyf powkzk}f1

:9

7/23/2019 500 Integrals


Wfgz# 9#79

$ + $ + $ +⎦⎦⎭

⎪

⎩⎩⎧

⎨∙∙2

Η∙

2

∘

∥ ∙

2

∙

∙

8

0 (

(8, η

η

ι

η

ι

ι

ι

η

ι

ι η

e

e c tc

tb

e

t

c

f

c

t

jtf t

c

c

• ^ykzkcn wvbbhzkoc kc yf}fywf oyjfy, hcj wkbplkm{kcn

$ +$ +

$ +$ +

$ +

$ +$ +

$ + $ +$ +

$ +$ +$ + $ +

⎦

⎦⎦⎦⎦

⎭

⎪

⎩

⎩⎩⎩⎩

⎧

⎨

∙%%

∙∙%

∙%

∙2

⎦⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩⎩

⎧

⎨

%%∙

%

∙%

∙∙∙

2

∙

∙∙

∙

∙∙

∙∙∙∙

∙

∙

∥

η

η

η η

ι

η η

η η η η

η

ι

ι

ι

η

ι

η η

ι

η

ι

η

ι

η

ι

η

ι

ι

η

(898

8

(0

8

(7

(9(8(8

7

7

9

98

77

9988

E

E

c

cc

t

c

cc

t

tb

t

t t

c

f

t

t t

c

f

jtf t

c

c

c

$ +$ +

$ +⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙2 ∘

∙

2%

∙∙∙ 8

08

8

(8

(8η η ι

ι η

η

e e

e c t t

e c

fc

i{ kcwpfgzkoc, cozkcn

yflhzkoc,

$ +$ +

$ +$ +$ + $ +

#0moyvcjfywzooj kw8om }hvlfzdf~dfyf

,798(8

(8 0

2

∙∙∙∙2∙∙

∙ 3

e

e e

e

η η η η η

η E

COZF1 cf~ }fywkoc zyhcwlhzfw boyf yfhjkl{ kczo

ghlgvlhzkocw moy ,:,7,9,82η Moybvlhw 9#77#89 ‗ 9#77#86#

COZF1 cf~ }fywkoc kw fhwkl{ jfyk}fj myob hio}f wolvzkoc,

∥ ,jtf tc

htb i{ wfzzkcn , ι ∙2h zo nk}f,

$ +$ +

$ +$ +$ +

$ +$ +

$ +$ + ⎦

⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙

2⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙∙

∙∙2

∘

∘∥

∙

2%

∙∙∙

∙

2%

∙∙∙∙

8

08

8

8

08

8

(8

(8

(8

(88

η η ι

η η ι ι

ι η

η

ι η

η

e e

e c t

e e

e ce

t tb

t

e c

f

t

e c

fjtf t

c

c

c

~dfyf $ +e 8∙ ghcgflw hcj cfnhzk}f wknc kw zhefc ovzwkjf#

COZF1 Moybvlh vwfmvl moy ~ykzkcn zdf lon"ftphcwkocw kc

Wfgzkoc 9#> ~dkgd ghc if zyhcwmoybfj myob lon# mvcgzkoc zo

ftpocfczkhl mvcgzkoc om zdf moybc

tbf t ι ∙5 f#n#, wff Wfgz#

9#>, iflo~, moy cf~ moybvlh,

$ + $ + ,lc

8∥ ∥ ∙∙2 jwf w t

jt t wbc

b

c

i{ gdhcnf om }hykhilf

$ +⎟ ⎠

⎓⎜⎒

⎕ 2222 ∙∙∙ wbb w f t tf

t

jtjw t w 88,,,lc #

:7

7/23/2019 500 Integrals


Wfgz# 9#79

COZF1$ + $ + ,lc

8∥ ∥ ∙∙2 jwf w t

jt t wbc

b

c

kw wol}fj iflo~ myob

fxvk}hlfcz ftpyfwwkoc,

$ + $ + #lc

8∥ ∥ ∙2 jwf w t

jt t wbc

b

c

Moybvlh 9#77#89

⎦⎭

⎪⎩⎧

⎨2

%2∙2

∙∙

∥ 88

c

b

c

fjtf t

c

c t

tb η ι

ι ι

Moybvlh 9#77#87

⎦⎭

⎪⎩⎧

⎨2

%2⎟⎟

⎠

⎓⎜⎜⎒

⎕ %∙2

∙∙

∥ 98

8

9 c

b t

c

fjtf t

c t tb

c

c

η ι ι

ι ι

Moybvlh 9#77#8:

⎦⎭

⎪⎩⎧

⎨2

%2

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ %%∙2

∙∙

∥

78

99

79

9

c

b

t t

c

fjtf t

cc t tb

c

c

η

ι ι ι

ι ι

Moybvlh 9#77#86

⎦⎭

⎪⎩⎧

⎨2

%2

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ %%%∙2

∙∙

∥

:8

==7

:79

97

c

b

t t t

c

fjtf t

ccc t tb

c

c

η

ι ι ι ι

ι ι

• Wviwzkzvzf }hlvfw om kczfnfy η kczo Moybvlh 9#77#88• Ftfygkwfw nk}fc iflo~, zdkw wfgzkoc#

Moybvlh 9#77#8=

U R tjtf t ι ι

π ι fym

9

8

9

∥ 2∙ • Gdhcnf om }hykhilf, tz ι 2 , hcj jfmkckzkoc om

$ + ∥ ∙2

t

z jz f t0

99fym

π

o Wfz $ +z f z fym 9 9

2∥ ∙

π hcj hppl{ lkbkzw, cozkcn zdhz

$ + #00fym 2

• Wff Moyb# 9#77#9 moy wkbklhy ghwf

∥ ∥ ⎦⎦⎭

⎪

⎩⎩⎧

⎨%∙

∙2

∙∙jt

t

fch

t

f

bjt

t

fcb

ht

b

ht

b

ht ccc

88

8 Vwf Kczfnyhzkoc i{ Phyzw#

$ +

∥ ∥

∥

⎦⎦⎭

⎪

⎩⎩⎧

⎨%∙

∙2∙∺

∙∙222

22

∙∙

∙

∙∙

∙

jt t

fch

t

f

b}jvv}

tbjt t}jt tj}

jtfchtjvfv

cb

ht

b

ht

b

bb

htcht

cc

cc

8

8

8

8

8

8

8 ,

,Lfz

,8Km 1Gdfge 2c zdfc Moybvlh 9#79:#8 nk}fw,

∥ ∥ ⎦⎭

⎪⎩⎧

⎨%∙

∙22

∙∙jt

t

fh

t

f

bjt

t

f K

b

ht

b

ht

b

ht

888

8

Moybvlh 9#796#6

∥ ∥ ⎦⎦⎭

⎪

⎩⎩⎧

⎨¸∙

∙2

∙

¸

∙

¸

jtcb

ht

b

ht

b

ht

t

fch

t

f

bjt

t

fccc

88

8

Whbf hw hio}f moy ∥ ∙

jtf

b

htc

hcj wkncw hwwkncfj#

Moybvlh 9#796#= • Wff Wfgz# 9#78, hio}f, moy jfyk}hzkoc5 zdfc hppl{

jfmkckzkoc om kcgobplfzf nhbbh mvcgzkoc

::

7/23/2019 500 Integrals


Wfgz# 9#79

$ + $ +

$ +c

b q jz f

z c

h

c

ht q hjt

t

f

z

ht

q

q q

c q q

b

ht

c

c

888

,8

8

8

8

∙2

∙2

∙∙Η∙2

∙∓

∙

%

%

%

∥

∥

• $ + mvcgzkoc,nhbbhkcgobplfzfkw,••Η

$ + t,hwnk}fc#9,4#760 Moybvlh θ Η

• ]fykm{ wolvzkoc i{ jkmmfyfczkhzkcn vcjfy zdf kczfnyhl

wknc

Moybvlh 9#796#4

$ +

$ +

⎦⎦

⎭

⎪

⎩⎩

⎧

⎨

2

2∙

2≠≠

%

⎦⎭

⎪⎩⎧

⎨ ∙∙∙

2

∘

∥ ∙

2∙

,###7,9

,###,9,88

,0,0

(

Fk

(

(88

0+$

b

c

b q hc

q c

hth

t

h

q

e q

c

f

jt t

f

c q

q

e e q c

e ht

b

ht

c

c

• Kbpyo}fj Moybvlh 9#796#4 moy kczfnfy q #

• #7,4#769Moybvlh myobJfyk}fj hw nk}fc kc NY = moy

yfhl t,

$ +$ +

$ + $ + ⎦⎭

⎪⎩⎧

⎨∙∙Η

∙2∙Η ∘

∙

2%

∙8

08

(8,0

(

8,

c

e e

e t

c

t

e f t

c tc

• Yfwzhzf kczfnyhl hw kcgobplfzf nhbbh mvcgzkoc, hw

pyf}kovwl{ jkwglowfj kc Moyb# 9#796#=1

$ + $ +c

b q

c

ht q hjt

t

f c q q

b

htc

8 ,

,88 ∙

2∙∙Η∙

2∥ %

• Plvn kczo1

$ +$ +

$ + $ + ⎦

⎭

⎪⎩

⎧

⎨∙∙Η

∙2∙Η ∘

∙

2

%

∙8

0

8

(8,0

(

8,

c

e

e

e t

c

t

e f t

c

tc , wviwzkzvzkcn

,,8 cht t

c

b" q c ∙222

hcj cozkcn zdhz $ + $ + t t Fk,0 ∙2∙Η zo nk}f1

$ + $ +$ +

$ +$ +

$ + $ + $ + $ +

$ + $ + $ +

$ +$ +

$ + $ +c q q

e e ce

ht q

q

e e ce e

e

ht q q q

c q q q

q

e e c

e ht

c

q q q

b

ht

ht q c

h

h

e f

q c

h

th

e f

q c

h

q chth

ht

e f

ht

q c

h

jt t

f

c

c

c

c

Fk(

(

(

+8$8

(8

(

88

(Fk888

(8

,0

(

88

8

088

8

088

8

8

8

08

8

%∙2

∙∙

∙∙∙

∙∙∙∙2

⎡⎢

⎡⎬

⎥

⎡⎫

⎡⎤

⎮

⎦⎦⎦

⎭

⎪

⎩⎩⎩

⎧

⎨

∙∙

∙∙Η∙∙

2

∘

∘

∘

∥

∙

2%%

∙

2%%

%

%

∙

2%

%

• ^ykzf wvbbhzkoc zfyb kc yf}fywf oyjfy hcj

wkbplkm{,

:6

7/23/2019 500 Integrals


Wfgz# 9#79

$ + $ +$ +

$ +$ +

$ + $ +

$ +

$ +

$ +

⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩

⎧

⎨

%%

∙%

∙%

∙

∙2

⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩

⎧

⎨

%%

∙%

∙%

∙

∙

∙

∙∙

∙∙∙∙

c

q

q c q ccq ht

c

q c q q c q cq q ht

q

t q

h t q

h q

t q

h q

t q

q

cf

ht

th

q

th

q

th

q

f q c

h

c

c

(

(

(7

(

(9

(

(8

8

(7(9(8

(

8

9

9

8

9988

E

E

$ + $ +

$ +$ + $ +

$ +$ +

$ +$ +$ + $ +

#98

8

(

(8

yflhzkoc,mhgzoykhlzdfcozkcn,kcwpfgzkoci{

(

(8

(98

8

8

8

0

8

9

9

8

e q q q q q

e q

t

h

q

e q

c

f

t q

h

t q q q

h

t q q

h

qt

c

f

q

e e q c

e ht

c

q

q c

q ccq ht

c

c

∙∙∙2

∙∙

⎦⎭

⎪

⎩⎧

⎨ ∙∙∙2

⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩

⎧

⎨

%%∙∙

%∙

%

∙2

∘

∙

2 ∙

∙

∙

∙

E

E


ghlgvlhzkocw moy ,7,9,82 q Moybvlhw 9#796#; ‗ 9#796#88#

COZF1 Moybvlh 9#79:#9 bh{ if yfwzhzfj myob hio}f

kczfnyhl ftphcwkoc i{ lfzzkcn

1nk}fzo,8,8, ∙222 c q ccb

$ +$ + $ + $ +htc

h

t

h

c

e c

f t

f cc

e e c

e ht

c

ht

Fk(8(8

(9 89

08 ∙%⎦⎭

⎪

⎩⎧

⎨

∙

∙∙

∙2

∙∙

2∙∙

∘∥

~dkgd jkmmfyw ocl{ kc zdf wzhyz/fcj wvbbhzkoc lkbkzw hcj

wzhzfbfcz om mhgzoykhl zfybw#

COZF1 Moybvlh 9#79:#9, ∥ ,jt t

fc

ht

kw h wpfgkhl ghwf om

c

b q jt

t

fc

htb

8,

∙2∥ , ~kzd #8,,8 ∙2↙2 b q cbc

Moybvlh 9#796#>

$ +

Fk

c

htjt

t

f chtc

2

∥

Jfyk}fj myob,$ +

c

ht

c

htjt

t

f cchtc +$Fk,02

∙Η∙2∥ ,

kc Moyb# 4#76;#8, ~dfyf 08 2∙2c

b q #

Wff Bhzdfbhzkghl Wvbbhy{ hcj Wfgz# 4#989, iflo~,

moy jfmkckzkocw om $ + t¸¸ Fk kc zfybw om kcgobplfzf

nhbbh mvcgzkoc, $ +#, tθ Η

Moybvlh 9#796#; Wviwzkzvzf }hlvfw om kczfnfy q kc 9#796#=

:=

7/23/2019 500 Integrals


Wfgz# 9#79

⎦⎭

⎪⎩⎧

⎨22

%∙

2∥

88"

+$Fk

c

b q

c

hth

ct

fjt

t

f c

c

ht

b

ht cc

Moybvlh 9#796#80

⎦⎭

⎪⎩⎧

⎨22

%∙∙

2∥

98"

9

+$Fk

99

9

9

c

b q

c

hth

ct

hf

ct

fjt

t

f c

c

ht

c

ht

b

ht ccc

Moybvlh 9#796#88

⎦⎭

⎪⎩⎧

⎨22

%∙∙∙

2∥

78"

=

+$Fk

==7

79

97

c

b q

c

hth

ct

fh

ct

hf

ct

f

jt t

f

c

c

ht

c

ht

c

ht

b

ht

ccc

c

Moybvlh 9#796#89

U R thh t

fjt

f htht

fymk

99

9π %∙2∥

Yfjvgzkoc moybvlh 9#796#6 hcj hppl{ 9#77#9,

∥ #9

jtfht

Moybvlh 9#77#8;

$ +

c

b q

z

f

c

c

t q jt

t

f

c

c

t

q

z q

c q

b

t

8

,

8

∙2∙2

∙Η∙2

∥

∥ ∓

%

∙

∙

ι

ι

ι

ι ι

• Wff Wfgz# 9#78 moy jfyk}hzkoc5 zdfc hppl{ jfmkckzkoc om

kcgobplfzf nhbbh mvcgzkoco Wfz bb ∙↙ kc ∥

∙ jtf tc

tb ι

• $ +$ + t,hwnk}fc,4;0p#4#760#9, Moybvlh

mvcgzkoc,nhbbhkcgobplfzfkw,

θ Η

••Η

• Vwfmvl moybvlh moy wolvzkocw kc 7#7"7#:, iflo~#

• ]fykm{ wolvzkoc i{ jkmmfyfczkhzkcn vcjfy zdf kczfnyhl

wknc

• }fykmkfj bhzdfbhzkgh#gob

Moybvlh 9#77#8; • Kbpyo}fj Moybvlh 9#77#8; moy kczfnfy q

• #7,4#769Moybvlh myobJfyk}fj hw nk}fc kc NY = moy

yfhl t,

$ +$ +

$ + $ + ⎦⎭

⎪⎩⎧

⎨∙∙Η

∙2∙Η ∘

∙

2%

∙8

08

(8,0

(

8,

c

e e

e t

c

t

e f t

c tc

• Yfwzhzf kczfnyhl hw kcgobplfzf nhbbh mvcgzkoc, hw

pyf}kovwl{ jkwglowfj kc 9#77#8>1

$ +c

b q

c

t q jt

t

f c q

b

tc

8 ,

, ∙2

∙Η∙2∥

∙ ι ι ι

• Plvn kczo1

:>

7/23/2019 500 Integrals


Wfgz# 9#79

$ +

$ +$ +

$ +$ +

$ + $ +

⎦⎦

⎭

⎪

⎩⎩

⎧

⎨

≠≠

22∙

2

∙∙

%⎦⎭

⎪⎩⎧

⎨ ∙∙∙

∙∙

2∙

∘

∥ ∙

2∙

0,0

,###,7,9 ,###,9,88

Fk(8

(

(88

ftp

ftp

8

0

ι

ι

ι

ι ι

ι

c

bc

b q

tcq

t q

e q

c

t

jt t

t

c q

q

q

e e q c

e e

c

b

c

$ +$ +

$ + $ + ⎦⎭

⎪⎩⎧

⎨∙∙Η

∙2∙Η ∘

∙

2%

∙8

08

(8,0

(

8,

c

e e

e t

c

t

e f t

c tc ,

wviwzkzvzkcn c ι t t ,c

b" q c 222

8, hcj cozkcn zdhz

$ + $ + t t ∙∙2Η Fk,0 , zo nk}f1

$ + $ + $ +

$ +

$ + $ +

$ +$ + $ +

$ + $ + $ + $ +∘

∘

∘

∥

∙

2%%∙

∙

2%%

∙

∙

2%

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• ^ykzf wvbbhzkoc zfyb kc yf}fywf oyjfy hcj wkbplkm{,

cozkcn zdhz kzw mkywz zfyb kc owgkllhzkcn wfykfw kw cfnhzk}f,

$ +$ + $ + $ + $ +

$ +

$ + $ + $ + $ +$ +

$ +

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ghlgvlhzkocw moy ,7,9,82 q Moybvlhw 9#77#90 ‗ 9#77#99#

COZF1 fhwkl{ jfyk}fj myob hio}f ftphcwkoc, ∥ ,jt t

fb

htc

i{

wfzzkcn , ι ∙↙h hcj yfhyyhcnkcn, ~dkgd hjjw h

$ +e 8∙ bvlzkplkfy zo wvbbhzkoc w{biol#

Moybvlh 9#77#84

$ +c

tjt

t

f c tc

ι ι ∙2∥

∙ Fk

Wff 9#796#> moy wkbklhy ghwf

]fykm{ wolvzkoc i{ jkmmfyfczkhzkcn vcjfy zdf kczfnyhl

wknc $goc}fyz $ +•Fk zo jfmkckzf kczfnyhl#

:4

7/23/2019 500 Integrals


Wfgz# 9#79

Moybvlh 9#77#90

$ +

⎦⎭

⎪⎩⎧

⎨2

∙2

∙∙∙2∥

∙∙

88

Fk

c

b q

c

t

ct

fjt

t

f c

c

t

b

tcc

ι ι ι ι

Moybvlh 9#77#98

$ +

⎦⎭

⎪⎩⎧

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∙2

∙

%%∙2∙∙∙

∥

98

9

Fk

999

9

c

b q

c

t

ct

f

ct

fjt

t

f

c

c

t

c

t

b

tccc

ι ι

ι ι ι ι

Moybvlh 9#77#99

$ +

⎦⎭

⎪⎩⎧

⎨2

∙2

∙∙

∙%∙2

∙

∙∙∙

∥

78

=

Fk

=

=779

97

c

b q

c

t

ct

fct

f

ct

fjt

t

f

c

c

t

c

t

c

t

b

t

c

ccc

ι ι ι

ι

ι

ι ι ι

Wviwzkzvzf }hlvfw moy kczfnfy q kc 9#77#8;

Moybvlh 9#77#97

U R t t

fjt

t

f t t

ι ιπ ι ι

fym

99

9∙∙2

∙∙

∥

Kczfnyhzkoc i{ phyzw $Moyb# 9#796#6+, gdhcnf om }hykhilf oc

wfgocj zfyb, hcj jfmkckzkoc om $ + tfym , Moyb# 4#960#8,

∥ ∙

t

z jz f0

99

π #

Moybvlh 9#798#9

$ +$ +

(

(8

80

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η

η η η

8,,8wfz %2↙2 ccbc η hcj wviwzkzvzf#

COZF1 pyopowfj moybvlh kw fwwfczkhll{ zdf whbf hw

wvnnfwzfj kc oclkcf NY = fyyhzh, hcj pvilkwdfj kc NY >,

dzzp1//~~~#bhzdzhilf#gob/fyyhzh/ny=_fyyhzh#pjm ,

Fyyhzh ':>, i{ Jyw# Boll hcj Ioyow‚ ftgfpz ~f wzhzf

$ +((

e c

c

∙moy zdf fxvk}hlfcz ftpyfwwkoc,

$ + ,

((

(((

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$ +$ +$ + $ +#898

(

( zdhz,cozkcn %∙∙∙2

∙e cccc

e c

cE

:;

7/23/2019 500 Integrals


Wfgz# 9#79

Moybvlh 9#79:#9

$ +$ +

$ +$ + U R0 Fk

(8

(8

(9

8

9

08

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∥

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h

t

h

c

e cf

jt t

f

c

c

e e c

e ht

c

ht

• Hlzfychzk}f ftpyfwwkoc moy Moybvlh 9#79:#91

Wfz 8,,8 ∙2↙2 c q cbc kc hio}f kczfnyhl,

$ + $ +

⎦⎭

⎪⎩⎧

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∙2

%⎦⎭

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Fk

(

(88

0+$

b

c

b q

q c

hth

t

h

q

e q

c

fjt

t

f c q

q

e e q c

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$ +

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2%

∙∙∙

c

e e

e c t tc t

e c

cfjtf t

ι

ι ι

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8

9 ⎟⎟ ⎠

⎓⎜⎜⎒

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∥ ι ι

tfjt tf t t ι ι

U R0≠ ι

U R0

99

79

99

≠

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⎓⎜⎜⎒

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∥

ι

ι

ι ι

ι

t

ι

tfjtf t t t

==7:79

977

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∥ ι ι

ι ι t

ι

t

ι

tfjtf t t t

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• $ +

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e e

e c t tc t

e c

cfjtf t

ι

ι ι kw jfyk}fj myob

nfcfyhl ghwf,

$ +$ +

$ +

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%

2

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8,9,###

8

(8

(88

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8

c

b

t

e c

fjtf t

e e

e c t tb

c

c

η

ι η

η η η ι ι

i{ wviwzkzvzkcn 8,,8 %2↙2 ccbc η

• ∥ ∥ ∥ ∙∙∙ jtf tjtf tjt tf t t t ι ι ι 79 -, oizhkcfj i{ wfzzkcn c

2 8, 9, 7, yfwpfgzk}fl{, kc

$ + #

(

(

08 ⎦

⎭

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∙∙2 ∘∥

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c

e e

e c t tc t

e c

cfjtf t

ι

ι ι

$ + t t

jtf t

ι ι

∙2∥ ∙

Fk

U R0≠

ι

Myob$ +c

t

t

jtf c tc

ι ι ∙2∥

∙ Fk ~kzd c 2 8#

$ +$ +

$ +

$ +$ +

$ +U R 0

(8

Fk8

(8

(98

88

9

08

≠∙

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⎪⎩⎧

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∙

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2

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∙

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∙

∙

∘

∥

ι ι

ι

ι ι

ι

c

t

tc

e cf

t

jtf

cc

c

e e c

e e t

c

t

$ + t

t

f

t

jtf t t

ι ι ι ι

∙∙∙

2∙∙

∥ Fk9

U R0≠ ι

∥ 2∙

7 t

jtf t ι $ +9

Fk

99

9

9

t

t

f

t

f t t ι ι

ι ι ι ∙%%

∙ ∙∙

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∥ 2∙

: t

jtf t ι

• $ +

$ +$ +

$ +$ +

$ +

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Fk8

(8

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8

9

08

∙

∙∙

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⎪⎩⎧

⎨

∙

∙∙∙∙2

∙

∙

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∙∙

∥ ∘

c

t

tc

e cfjt

t

f

c q c

c

e e c

e e t

c

t

ι ι

ι ι ι

kw jfyk}fj myob nfcfyhl ghwf,

$ +$ +

$ +

$ +

$ +⎦⎭

⎪

⎩⎧

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∙2

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∙

2∙

∙∙

,###7,9 ,###,9,8

8

(

Fk

8

(

(88

8

0

bc

b

q q c

t

t q

e q

c

fjt

t

f

c q q

q

e e q c

e e

t

b

tcc

ι

ι

ι ι ι

i{ wviwzkzvzkcn 8,,8 ∙2↙2 c q cbc

• ∥ ∥ ∥ ∙∙∙

:79-,

t

jtf

t

jtf

t

jtf t t t ι ι ι

1 wfz c 2 9, 7, :,

yfwpfgzk}fl{ kc #∥ ∙

jt t

fc

t ι

60

7/23/2019 500 Integrals


Wfgz# 9#79

$ +=

Fk

==7

79

97

t

t

f

t

f

t

f t t t ι ι

ι ι ι ι ι ∙∙∙%

∙ ∙∙∙

U R0≠ ι ∥ jt t c

H ihwkg kczfnyhl fxvhl zo $ +#88

8

∙≠%

%

cc

tc

Wdo~ jfyk}hzkoc

vwkcn ftpocfczkhl mvcgzkocw#Ocf hppyohgd1

Kc kczfnyhl, ∥ 2 jt t K c , wfz tcc f t lc2 myob jfmkckzkoc#

Zdfc, ∥ 2 jtf K tc lc# Vwf gdhcnf om }hykhilf,

tf t

jtjw t w w 222 ,,lc , zo nk}f,

$ +∥

%2 jwf K wc 8lfhjkcn zo ∥ %

2 jz c

f K

z

8i{ h wfgocj

gdhcnf om }hykhilf, $ + wcz 8%2 , ~dkgd lfhjw zo hcw~fy,

$ +

8888

8lc+8$8

%2

%2

%2

%2

%%%

c t

cf

cf

cf K

c tc wcz

i{ ihge wviwzkzvzkoc hcj lh~w om ftpocfczw#

COZF1 Ozdfy hppyohgdfw hyf powwkilf ocgf$ +

∥ %2 jwf K wc 8

kw oizhkcfj#

∥ jtc

8

Ftfygkwf# Wkbklhy zo ∥ jt tc #

∥ jt t9

8- ∥ jt

t7

8Ftfygkwf# Gdfge hnhkcwz wolvzkoc hio}f moy ∥ jt

tc

8#

∥ jt t t

Gobpvzfyw ghccoz wol}f zdkw kcjfmkckzf kczfnyhl kc glowfj

moyb# Zy{ kz oc bhzdfbhzkgh#gob# ^f ghc ~ykzf, myobjfmkckzkoc,

$ + #lclc t t t t t ff t 22 Hcj zdf jfyk}hzk}f kw nk}fc i{ zdf

gdhkc yvlf,

$ +$ +

$ +#8lc

8lclc lclc

lc

%2

%222

t t

f tfjt

t tj

jt

fj

jt

jt

t

t t t t t t t

Wviwzkzvzf hcj wff yfwvlz# Co flfbfczhy{ moybw ghc if

wviwzkzvzfj zo pyo}kjf h glowfj moyb wolvzkoc#

Wff NY, Wfgzkoc 8#988, moy wfykfw ftphcwkoc om t t i{

looekcn hz th hcj wfzzkcn # th ↙

$ +

c

b

jz z ch

jtf t

cht

cf

htb

8

8lc

8

0

88

%2

⎟ ⎠

⎓⎜⎒

⎕ ∙2∥ ∥

∙∙

η

η

η

η

Jfyk}fj i{ hio}f moybvlh $~kzd +wfz zo wz ,

$ + $ +

$ +#

8,

8

,8

8

8

8

c

bjwf w

ch

ch

htjtf t

w

ht

chtb

c

c

%2

∙2

∙Η∙2

∙∓

∙

∙∙

∙

∥

∥

η

η

η

η

η

η

η

Vwf gdhcnf om }hykhilf oc jfmkckzf kczfnyhl,

68

7/23/2019 500 Integrals


Wfgz# 9#79

$ +jwf w

ch

w

htc

∙∓

∙

∙∙

∥ ∙ 8

88 η

η

η

1

z z w wz

z fz

zjwjwfjz fz c

c

ht

ht

w w

8lclc,lc

,0,

,

2∙2∙2

22

∙2∙22

∓∙

∙∙

Zdfc,

$ +$ +

$ +jz

z ch

jz z

z z

ch K

cht

cht

f

f

∥

∥

∙∙

∙∙

⎟ ⎠

⎓⎜⎒

⎕ ∙2

∙∙

∙2

0

88

08

8

8lc

8

lc8

η

η

η

η

η

η

Zdvw,

$ + $ +

$ + $ +

#8

,8

lc88

,8

0

88

8

8

8

c

b

jz z ch

jwf wch

ch

htjtf t

cht

c

c

f

w

ht

chtb

%2

⎟ ⎠

⎓⎜⎒

⎕ ∙2

∙

2∙Η∙

2

∥ ∥

∥ ∙∙

∙∓

∙

∙∙

∙

η

η

η

η

η

η

η

η

η

η

• Jfyk}f wkbklhy lonhykzdbkg ftpyfwwkocw moy zdfwf 7

hlnfiyhkg"ftpocfczkhl kczfnyhlw nk}fc fhylkfy1

∥ ∥ ∙

∙ jt t

fjt

fjtf t

b

t

b

ht tb

cc

c ι

ι --

]fykm{ fhgd wolvzkoc i{ jkmm# vcjfy zdf kczfnyhl

$ +

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ %2

∙%2%∙

∙

∥

99lc9

8lc9

8

8

htht

ht

ht

ht

ffh

tfh

jtf

f

^f wdo~ z~o hlzfychzk}f wolvzkocw#

• Wfphyhzf zdf zfybw,

∥ ∥ ∥ %∙

%2

%∙

jtf

jtf

fjt

f

fhtht

ht

ht

ht

8

8

88

8


hwjtjthfjwf w htht 222 ,

Zdfc,

$ + $ + $ +⎦⎭

⎪⎩⎧

⎨

%∙

%2

%∙

%2 ∥ ∥ ∥ ∥ 88

8

8

8

8 w w

jw

w

jw

hhw

jw

whw

jw

w

w K

Vwf phyzkhl myhgzkocw moy wfgocj kczfnyhl,

$ + 8

88

8

8

%∙2

% w w w w, zo nk}f,

$ + $ +U R $ +U R w wh

w w wh

jw w w w

jw

h K

lc8lc98

8lclc8lc8

8

88

8

8

∙%2%%∙%2

⎦⎭

⎪⎩⎧

⎨⎟ ⎠

⎓⎜⎒

⎕

%∙∙

%2 ∥ ∥

69

7/23/2019 500 Integrals


Wfgz# 9#79

• Hz zdkw wzhnf z~o wolvzkocw ghc if oizhkcfj# Mkywz

wolvzkoc kw oizhkcfj i{ wviwzkzvzkcn htf 1

$ +U R

$ +U R

$ +U R $ + t

h

fhtf

h

ffh

w wh

K

htht

htht

∙%

2∙%2

∙%2

∙%2

8lc98lc9

8

lc8lc98

lc8lc98

Zdkw kw gobpvzfy wolvzkoc hz bhzdfbhzkgh#gob

Hlwo, vwkcn zdf lon# yflhzkoc kc Bhzd# Wvbbhy{,

,lclclc⎟⎟⎟

⎠

⎓

⎜⎜⎜

⎒

⎕

2∙h

i

{

th {i th ~kzd ,8,9 22 ih

zdf wfgocj wolvzkoc kw1

$ +U R

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ %2⎟

⎟ ⎠

⎓⎜⎜⎒

⎕ %2

⎟⎟

⎟

⎠

⎓

⎜⎜

⎜

⎒

⎕ %

2∙%2

∙∙999

8

9

8

9

8

lc9

lc9

8

lc

9

lc8lc9

8

htht

ffh

w wh

w

w

h w wh K

Gkzfj kc Pyvjckeo}, fz hl#, ]ol# 8, Moybvlh 8#7#8#>, p#

87>

Ftfygkwf1 Wdo~ ⎟⎟ ⎠

⎓⎜⎜⎒

⎕ ∙2

∙% ∙

∥ 99lc9

8

8htht

ht

ht

ffh

jtf

f

⎟⎟ ⎠

⎓

⎜⎜⎒

⎕

%

%

2%

∙ %∙

%

∥ cb

htb

cb

cht

ht

ht

cffch

cb

jtcf

bf

lc

Ocf wolvzkoc wol}fj i{ wfphyhzkoc om zfybw, gdhcnf om

}hykhilf, ,ht

f w 2 phyzkhl myhgzkoc,

$ + $ +h thhth t t ¸¸2

¸

888b , hcj yflhzkoc,

⎟⎟⎟

⎠

⎓

⎜⎜⎜

⎒

⎕

2∙h

i

{

th {i th lclclc #

Gobpvzfy pyo}kjfw hlzfychzk}f wolvzkoc,

$ + $ + #lcc

btcf

ch

cbjt

cf

bf ht

ht

ht

∙%%

2%

∙∥

Gdfge hnhkcwz kczfnyhl wolvzkoc, jtff

ht

ht

∥ %∙88

~kzd ,822 cb hcj i{ jkmmfyfczkhzkoc#

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ ∙

%2

∙

%%

∙%∥ cb

htb

cb

cht

ht

ht

cffch

cbjt

cf

bflc

$ +U Rhtbbcht

ht

ht

fcfch

jtcf

bf ∙%2

%%

∥ lc8

Ftfygkwfw#

Wkbklhy zo hio}f kczfnyhl,

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ %

%2

%∙

%∙

%∥ cb

htb

cb

cht

ht

ht

cffch

cbjt

cf

bflc

• Fhgd kczfnyhl dhw z~o hlzfychzk}f wolvzkocw# ^f nk}f

67

7/23/2019 500 Integrals


Wfgz# 9#79

$ +U Rhtbbcht

ht

ht

fcfch

jtcf

bf ∙∙2

∙∙

∥ lc8

ocl{ ocf wolvzkoc, ihwfj oc zdf lon# yflhzkocw,

$ +

⎟

⎟⎟

⎠

⎓

⎜

⎜⎜

⎒

⎕

2⎟⎟

⎠

⎓⎜⎜

⎒

⎕ 2∙

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ 22%

h

ii

h

h

i

ih

{

th

{

t {i th

t{h { t {i th

lclclclc

lclclclc

Zdf gobpvzfy nk}fw zdf wfgocj wolvzkoc#

∥ 2 t t fjtf

]fykm{ wolvzkoc zo zdkw ihwkg kczfnyhl i{ fhylkfy ftpocfczkhl

kczfnyhl moybvlh,

$ + $ + $ +

$ + iflo~#4#769,Wfgz#,hcj,8~kzd,8

8,8 8

88

c,"tc

b

jz fz chch

htjtf t z

ht

chtb

c

c

Η2%

2

∙2

∙Η∙2 ∙

∓

∙

∙∙∙

∥ ∥

η η

η η

η

η

η

η

∥ ∙∙ ∙2 t t fjtf

Wdo~ i{ Moybvlh 9#77#88, hio}f,

$ +

$ + iflo~#4#769,Wfgz#,,hcj,8~kzd,8

8, 8

tcc

b

jz fz cc

tjtf t z

t

c

tb

c

c

Η2%

2

∙2Η

∙2 ∙

∓

∙∙

∥ ∥

η η

ι ι ι η

ι

η η η

ι

$ +h

f t

f

jt ht

ht

8lc

8

%∙2

%∥

• Vwf gdhcnf om }hykhilf,

hwjtjthfjwf w htht 222 , • Zdfc, i{ phyzkhl myhgzkoc ftphcwkoc hcj yvlfw om

lonhykzdbw,

$ +

$ + $ +U R

$ + $ +U R $ +h

f tff

h

f

f

h w w

h

jw w wh w

jw

wh K

hththt

ht

ht

%∙2%∙2

⎟⎟ ⎠

⎓⎜⎜⎒

⎕

%2%∙2

⎟ ⎠

⎓⎜⎒

⎕

%∙2

%2 ∥ ∥

8lc8lclc

8

8lc

88lclc

8

8

888

8

88

$ +hihtfjti t

f hiht

%2%

∙

∥ Fk

• Vwf gdhcnf om }hykhilf,

i w tjtjwi t w ∙22%2 ,, • Zdfc,

$ +

$ +

$ +hihtf

hwfjw w

ffjw

w

f K

hi

hihw

hii wh

%2

222

∙

∙∙∙

∥ ∥ Fk

Fk

I{ Moybvlh 9#796#>,$ +

,Fk

c

htjt

t

f chtc

2∥ ~kzd #82c

$ +9

88

h

hhtfjt

f

t ht

ht

%%∙2

% ∙

∥

Ftfygkwf# Gdfge hcw~fy oc bhzdfbhzkgh#gob

$ + $ + t t

tf

tf tjt

f

t9

9

Lk9

8lon8

%∙∙2∙∥

• Vwfw h wpfgkhl mvcgzkoc, zdf jklonhykzdb, $ +,Lk9 t jfmkcfj

6:

7/23/2019 500 Integrals


Wfgz# 9#79

hw hc kcmkckzf wfykfw, $ + ∘∓

2

<28

99 #8,Lke

e

q e

q q Zdf

kcjfmkckzf kczfnyhl kw zhefc hw1 $ +$ +

#8lon

Lk9 jt t

t t ∥

∙∙2

• Kc ∥ ∙

2 jt

f

t K

t

8

, vwf gdhcnf om }hykhilf, , tf w 2 hcj

phyzkhl myhgzkoc ftphcwkoc zo nk}f,

∥ ∥ ∙∙

2 jw w

wjw

w

w K

lc

8

lc#

Zdf wfgocj kczfnyhl kw wol}fj i{ gdhcnf om }hykhilf,

,lc wz 2 zo nk}f,

$ +#

99

lc

9

lc 999 t wz zjz jw

w

w2222 ∥ ∥

Zdf mkywz kczfnyhl kw boyf kc}ol}fj# Zdf ftfygkwf iflo~

~kll wdo~ zdhz, hmzfy kczfnyhzkoc i{ phyzw,

$ + $ + w w wjw w

wjw w

w9Lk8lclc8

lc

8

lc

%∙2∙2∙ ∥ ∥ ,

wo zdhz,

∥ 2∙

jtf

t t 8

$ + $ +9

Lk8lclc9

9

t w w w ∙%∙

~dkgd lfhjw zo wolvzkoc ~dfc tf kw wviwzkzvzfj,

$ + $ + t t

tf

tf tjt

f

t9

9

Lk9

8lon8

%∙∙2∙∥

COZF1 Jo coz gocmvwf cozhzkoc om $ + t9Lk ~kzd lonhykzdb"

kczfnyhl mvcgzkoc, $ +#lk t

• Ftfygkwf1 jfyk}f wflfgzfj gobboc jklonhykzdbyflhzkocw1

66

7/23/2019 500 Integrals


Wfgz# 9#79

$ +$ + $ +

$ +

$ + $ +

$ +

$ +$ + $ +

$ +$ +

$ +

$ +$ +

$ +⎟

⎠

⎓⎜⎒

⎕

⎟ ⎠

⎓⎜⎒

⎕ ⎦⎭

⎪⎩⎧

⎨⎟

⎠

⎓⎜⎒

⎕

⎟ ⎠

⎓⎜⎒

⎕ ⎟

⎠

⎓⎜⎒

⎕

⎟ ⎠

⎓⎜⎒

⎕ ⎦⎭

⎪⎩⎧

⎨⎟

⎠

⎓⎜⎒

⎕

⎟ ⎠

⎓⎜⎒

⎕ ⎟

⎠

⎓⎜⎒

⎕ ⎦⎭

⎪⎩⎧

⎨⎟

⎠

⎓⎜⎒

⎕

⎟ ⎠ ⎓

⎜⎒ ⎕

⎟ ⎠

⎓⎜⎒

⎕

⎟ ⎠

⎓⎜⎒

⎕ ⎦⎭

⎪⎩⎧

⎨⎟

⎠

⎓⎜⎒

⎕

∥

⎟ ⎠

⎓⎜⎒

⎕ ⎟

⎠

⎓⎜⎒

⎕

⎟ ⎠ ⎓

⎜⎒ ⎕

∙2

∥ ∙∙∙∙2∙

∙2∙2

∥ ∙¸∙¸2¸

∙2∥ ∙∙∙∙2∙

∙2∙2

∥ ∙¸∙¸2¸

∙2∥

∙

%¸2∥ ¸

∙2∥ ∙2¸

it

hit

h

it

h

ithit t

t

jthit

it

it

h

it th

h

it

h

itith t

t

jtith

t

t

h

t

h

th t t

t

jth t

t

h t

h

t th

h

t

h

t th t

t

jt th

h t

tjt

th

tjt

h

t

h

t t

h t

tjt

t

t t

t

jt t

h

h

9Lk

9

lc

9

Lk8lclclclc

9

9

9

9

9

9

9

9

9

9

9

9

9

Lk

9

lcLklclc

Lk8lclclclc

Lk9

lcLk8lclclc

lc

9Lk9

lcLklclc

Lk8lclclclc

hio}fwff lclc

Lk8lclclc

8

9Lk

9

lc

9Lk

8lc

bb

b

bb

b

b

bb

Ft1$ +

∥ ∥ ∥

⎦⎭

⎪⎩⎧

⎨⎟⎟ ⎠

⎓⎜⎜⎒

⎕ ¸

2

⎦⎭

⎪⎩⎧

⎨⎟⎟ ⎠

⎓⎜⎜⎒

⎕ ¸

2¸

t

jt t

h t

t

jth

th

t

jt th8lc8lc

lc5 mkywz

kczfnyhl yfxvkyfw z~o kczfnyhzkocw i{ phyzw ~kzd phyzkhl myhgzkoc

kjfczkz{ oc jv zfyb5 wfgocj hcj zdkyj wolvzkocw vwf lon# yflhzkoc,

,lonlonlon { t t{ %2 hcj jfmkckzkoc om $ +#Lk9 t • COZF1 $ + t9Lk dhw vwfmvl yfwvlzw kbpoyzhcz moy jfmkckzf

kczfnyhlw5 f#n#, $ + $ +89

9

9Lk5=

9

9Lk 88π π

∙22 ∙ # Wff Zdf

ôlmyhb Mvcgzkoc Wkzf dzzp1//mvcgzkocw#~olmyhb#gob/ #

bt

btbt

ifh

jt

f

ih

jt

f

ih

jt

∙∙%∥ ∥ -- • Zdyff ftfygkwfw# Wff ∥ % btifh

jtjfyk}hzkoc kc zdf

Bhzdfbhzkghl Wvbbhy{#

• Gdfge wolvzkoc oc bhzdfbhzkgh#gob

∥ ∙ jtf t t9

7 ι

Jfyk}f myob Moybvlh 9#77#89, hio}f,

⎦⎭⎪⎩⎧⎨ 2%

2∥ ∙ 98

cbjtf t

c

tb η ι

∥ ∙ jtf t t ι 9

Jfyk}f myob Moybvlh 9#77#89, hio}f,

⎦⎭

⎪⎩⎧

⎨2

%2∥

∙ 78

c

bjtf t

c tb η ι

• Zy{ ozdfy gobikchzkocw om cb, moyc

b 8%2η 2 kczfnfy#

6=

7/23/2019 500 Integrals


Wfgz# 9#79

$ + $ + 0 ,lkFk <22∙2 ∥ ∥ ∓∙

∓

∙

∙

tfjz z

fjz

z

f t t

t z

t

z

Kc zdkw jfmkckzkoc, wdo~ fxvk}hlfcgf om z~o kczfnyhlw i{

wzhzkcn zdf kcjfmkckzf kczfnyhl moybw, wviwzkzvzkcn zdf

lkbkzw, hcj hppl{kcn pyopfyzkfw om zdf ftpocfczkhl"

kczfnyhl mvcgzkoc wvbbhykqfj kc Bhzdfbhzkghl

Wvbbhy{# Zdfc wdo~ $ + $ + ∥ 22

tf

t

z

jz f t

0lc

lkFk i{ gdhcnf

om }hykhilf, #lc z w 2

$ + 0 ,lkbFk0

3⎦⎭

⎪⎩⎧

⎨%∙2 ∥ ∥

∓ ∙∙

∙

∙

%↙ tjz

z

fjz

z

f t

z

t

z

ζ

ζ

ζ

Ghvgd{ Pykcgkphl ]hlvf $P]+ moy jfmkckzkoc om ftpocfczkhl"

kczfnyhl mvcgzkoc, 03 t #

• Wdo~ zdf lkbkz ghlgvlhzkoc i{ wzhzkcn zdf kcjfmkckzf

kczfnyhl moybw, wviwzkzvzkcn zdf lkbkzw, hppl{kcn

pyopfyzkfw om zdf ftpocfczkhl"kczfnyhl mvcgzkoc kc

Bhzdfbhzkghl Wvbbhy{, hcj f}hlvhzkcn zdf lkbkz,#0%↙ζ

$ + $ + 8,lcFklc

lk0

<22 ∥ t tz

jz t

t

Kc zdkw jfmkckzkoc, wdo~ zdf yflhzkoc, $ + $ + ,8,lcFklk <2 t t t

i{ gdhcnf om }hykhilf,

z w lc2 oc kczfnyhl, #lc

0∥

t

z jz

$ + $ +

8

,lcFklclc

lkblk8

8

00

3

2⎦⎭

⎪⎩⎧

⎨%2 ∥ ∥

%

∙

↙ t

z

jz

z

jz t

t

ζ

ζ

ζ

Ghvgd{ Pykcgkphl ]hlvf P] moy jfmkckzkoc om lonhykzdb"

kczfnyhl mvcgzkoc, $ + #8,lk 3 t t

• Pfymoyb whbf opfyhzkocw hw moy hio}f ftpocfczkhl"

kczfnyhl mvcgzkoc, $ + tFk

$ + t

fjt

t

tf t t

%2

%∥ 889

• Gdhcnf om }hykhilf,

jtjw t w 2%2 ,8

• $ +⎦⎭

⎪⎩⎧

⎨∙2

∙2 ∥ ∥ ∥

∙∙

jw w

fjw

w

ffjw

w

f w K

w w w

9

8

9

88

Kczfnyhzkoc i{ phyzw oc mkywz kczfnyhl ~kzd

w w f}jwfj} w

jwjv

wv 22∙22 ,5,

89

,

zo pyojvgf,

#88

888

99

8

%2

%2⎟⎟

⎠

⎓⎜⎜⎒

⎕ 2

⎦⎭

⎪⎩⎧

⎨∙%2

%∙∙

∙

∥ ∥

t

f

t

ff

w

ff

jw w

fjw

w

f

w

ff K

t t w

w w w

COZF1 Hj}hcgfj wolvzkoc# Kc mkywz kczfnyhl hio}f,

$ + wjw w

f wFk2∥ , myob jfmkckzkoc,

$ + wjw w

f w∙∙Η2∥ ,8

9i{ Moybvlh 9#796#>, hio}f,

$ + $ + $ +c

b q jz f

z c

h

c

ht q hjt

t

f z

ht

q

q q c q q

b

ht

c

c

8,

88,88

88∙

2∙

2∙∙Η∙

2 ∙∓

∙%

%%

∥ ∥

Zdfc, $ + $ + w

f w w

w

∙2∙∙Η Fk,8 i{ $ + t∙∙Η ,8 kc Wfgz# 4#769,

6>

7/23/2019 500 Integrals


Wfgz# 9#79

iflo~ wo zdhz,

$ +

$ + $ + #8

FkFk

8

88

9

8

9

8

%22⎦

⎭

⎪⎩⎧

⎨%∙2

⎦⎭

⎪⎩⎧

⎨∙2

∙2

∙∙

∙∙

∥ ∥ ∥

t

f

w

ff

w

f w wf

jw w

fjw

w

ffjw

w

f w K

t w w

w w w

Gkzfj J~kndz, 8;=8, KZFB 6>0

$ + $ +hth

fjt

ht

tf htht

%2

%∥ 8899

Ftfygkwf# Wkbklhy zo hio}f,$ +

jt t

tf t

∥ %9

8#

Gdfge wolvzkoc oc bhzdfbhzkgh#gob, hcj gobphyf ~kzd

hio}f kczfnyhl moy #82h

Gkzfj J~kndz, 8;=8, KZFB 6>0#8

∥ ∙

jtf

t

9

9

^f wdo~ zdhz zdkw kczfnyhl kw fxvhl zo π 9 moy lkbkzw #∓¸

Kz kw zdf ihwkw moy zdf Nhvwwkhc pyoihiklkz{ jfcwkz{

mvcgzkoc#

• Myob Moybvlh 9#77#8=, hio}f,

U R tjtf t ι ι π ι fym

98

9

∥ 2∙ # Wfz 982 ι hcj vwf pyopfyz{ moy

fyyoy mvcgzkoc, $ + 8¸2∓¸έ $wff Wfgz# 4#76;, iflo~+#

Zdvw,

$ + $ +U R

$ +U R $ + #9999

8+889

9

8

fym fym 99

8

9fym 9

9

8

9

9

8

π π π

π π

22∙∙2

∓∙∙∓%2∓∙

∓%⎦⎭

⎪⎩⎧

⎨⎟⎟ ⎠

⎓⎜⎜⎒

⎕ 2∥

%∓

∓∙

∙ tjtf

t

Gobphyf wolvzkoc ~kzd wzhcjhyj hppyohgd kc}ol}kcn h

jovilf kczfnyhl zyhcwmoybfj zo polhy gooyjkchzfw# Wff

Ifyw, Ghlgvlvw, ]ol# 9, p# ;94# Hlwo wdo~c kc iooew oc

pyoihiklkz{ zdfoy{#

COZF1 Wkcgf zdf kczfnyhl ∥ ∙

jtf

t

9

9

goczhkcw hc f}fc

mvcgzkoc, +,$+$ t m t m 2∙ $wff NY, Wfgz# 7#07+, zdfc,

∥ ∥ %∓

∙%∓

∓∙

∙

20

9

8

9

8 99

9 jtfjtf t t

#

Kc zdkw hppyohgd, mkywz wol}f zdf kcjfmkckzf kczfnyhl,

9

8

9

9

9

89

9,

9

89

⎟

⎠

⎓⎜

⎒

⎕

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ Η

∙2∥ ∙

t

jtf

t i{ Moybvlh 9#77#80,

$ +,

8,

8, 8

c

bjz fz

cc

tjtf t z

t

c tb

c

c %2∙2

Η∙2 ∙

∓∙∙

∥ ∥ η ι ι

ι η

ι

η

η η

ι

wo zdhz,

64

7/23/2019 500 Integrals


Wfgz# 9#79

π 9989,

980,

989

0

9

9,

9

89

99

9

0

9

8

9

8 99

2⎟ ⎠ ⎓⎜

⎒ ⎕ Η2⎦

⎭⎪⎩

⎧⎨ ⎟

⎠ ⎓⎜

⎒ ⎕ ∓Η∙⎟

⎠ ⎓⎜

⎒ ⎕ Η2

∓%

⎦⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩⎩

⎧

⎨⎟⎟

⎠

⎓⎜⎜⎒

⎕ Η

∙22 ∥ ∥ ∓%

∙∓%

∓∙

∙

t

jtfjtf t t

i{ zdf pyopfyzkfw om kcgobplfzf nhbbh mvcgzkocw, hcj vwf

om eco~c yflhzkoc, #9

8π 2⎟

⎠

⎓⎜⎒

⎕ Η

COZF1 Kc zdf kczfnyhl ∥ %∓

∓∙

∙,

9

9

8

jt tf t

zdf kczfnyhcj kw

gocwkjfyfj hc ojj mvcgzkoc, +,$+$ t m t m ∙2∙ $wff NY, Wfgz#

7#07+, wo zdhz

#09

9

8

2∥ %∓

∓∙

∙jt tf

t

Zdf kczfnyhl, ,09

89

9

8

2∥ %∓

∓∙

∙jt tf

t

π kw zdf “bfhc‒ om zdf

Nhvwwkhc pyoihiklkz{ jfcwkz{#

$ +htjt t

fht

Fk2∥

Wff wfykfw ftphcwkoc kc J~kndz, 8;=8, KZFB 6=4#8

COZF1 J~kndz iooe kw vwfmvl moy ozdfy wfykfw ftphcwkocw,

hcj goczhkcw wobf kcmoybhzkoc oc zdf nhbbh mvcgzkoc,

$ +# tΗ

0lkb 2%∓↙ t

c

t f t moy hll }hlvfw om c • Zdkw kw hc kbpoyzhcz lkbkz moy h mvcjhbfczhl

yflhzkoc om zdf nhbbh mvcgzkoc,

$ + $ +#80

t tjz fz t z t Η22%Η ∙∓

∥ Vwkcn kczfnyhzkoc i{ phyzw

~kzd ,, jz fj}z v z t ∙22 zdfc,

$ +

$ +

$ +#

0lkb

08

0

8

0

t t

t tf

z

jz fz tz

z

f

z jz fz t

z

t

z

z t

z

tz t

Η2

Η%⎦⎭

⎪⎩⎧

⎨∙∙2

%2

%∓2∙22%Η

%∓↙

∙∓

∙∙∓

∥ ∥

• Zdf lkbkz 0lkb 2%∓↙ z

t

z f

z mollo~w myob zdf zdfoyfb ~dkgd

kw jfyk}fj myob l.Dúpkzhl.w yvlf ⎟ ⎠

⎓⎜⎒

⎕

∓

∓ghwf i{ zhekcn

yfpfhzfj jfyk}hzk}fw om cvbfyhzoy hcj jfcobkchzoy $wff

Ifyw, Ghlgvlvw, pp# 6:> mm#+#

6;

7/23/2019 500 Integrals


Wfgz# 9#79

o Zy{ zo pyo}f zdhz 0lkb 2%∓↙ t

c

t f

t

i{ wdo~kcn zdhz zdf cvbfyhzoy f}fczvhll{ ifgobfw (c i{

yfpfhzfj jkmmfyfczkhzkoc# Moy fthbplf,

(897+7+$9+$8$

97+7+$9+$8$

+9+$8$

+8$

(7897977

(9899

7

98

97

9

ccccc

jt

jtcccc

jt

jtccc

jt

jtcc

jt

jtc

jt

jt

jt

jt

jt

jt

jt

jt

jt

jt

jt

jt

c

ccc

2⋆⋆∙∙∙2

⋆∙∙∙2

∙∙2

∙22

2⋆⋆2⋆22

2⋆22

∙

∙∙

E

E

Hppl{ zdkw yfwvlz zo #08

lkb(lkb 22 %∓↙%∓↙ z z z

t

z f t

fz

=0

7/23/2019 500 Integrals


=8

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚



Wfgz# 9#>88

$ +$ +∘∥

2

∙

∙∙2

b

e

e be b t

e b

b t tjt

0

lc(

(8lc

U R0≦b

• Hlzfychzk}f yfwzhzfbfcz om 9#>88 $9cj moybvlh+• Jfyk}fj hio}f kc Wfgz# 9#79 vcjfy Moybvlh

9#77#6,

$ +$ +

$ +$ +

⎦⎭

⎪⎩⎧

⎨≠≠2

%2

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙2∥ ∘ %

∙∙∙

2

0,08,9,###,8

(8

(88

8

88

0

hcc

b

h

t

e c

fjtf t

e

e c

e

e ht

htb

c

c

η

η

η η η

COZF1 vclkef 9#>88, km ,,0 t K b 22 h zyk}khl yfwvlz#

COZF oc cozhzkoc1 $ + bbb t t t lclclc ≠2

$ +

$ +$ +

$ +ithe b

b

i

ith

jtith

e bb

e

e

b

%∙

∙%

2%

∙

2∘

∥

lc(

(8

lc

0


$ + ,,,lc ithfjtith

ijwithww %2

%2%2 zo

nk}f,

∥ 22 jwfwi

K wb8

$ +$ +

e bb

e

e w

we b

b

i

f ∙

2 ∙∙∘

(

(8

0

,

i{ hio}f kczfnfy"ftphcwkoc Moybvlh kc Wfgz#9#79,

$ +$ +

$ +$ +

⎦⎭

⎪⎩⎧

⎨2

%2

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙2∥ ∘ %

∙∙∙

2

8,9,###8

(8

(88

8

88

0 c

b

h

t

e c

fjtf t

e

e c

e

e ht

htb

c

c

η η

η η η

~kzd phyhbfzfyw,

$ +U R8,8,,lc %2222%22 bchbbith tw η Yfjvgfw zo Moybvlh 9#>88 moy #8,0 22 ih

Wfgz# 9#>98

Moy 8∙2b

$ + $ +∥ 22 t t t

jtlklcFk

lc

Gdhcnf om }hykhilf1

$ + $ +∥ ∥ 2222

2222

t tjww

fjw

w

t K

tff tjt

jw tw

w

tw

lklcFk

#,8

,lc lc

i{ kczfnyhl pyf}kovwl{ wvibkzzfj, Moyb# 9#796#>,

$ +,

Fk

c

htjt

t

fcht

c

2

∥ hcj Moybvlh 4#9:0#8#

Wfgz# 9#>98

Moy 822 cb

∥ ∙2:

lc9

lc99

t t

t tjt t

H wzhcjhyj zftziooe kczfnyhl, myob

Moybvlh 9#>98#8 ~kzd #822 cb

=9

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

Wfgz# 9#>99

$ +$ + $ +

∘∥ 2

%

∙%

%∙∙2

b

e

e

e be cbc

c

t

e b

b t tjt t

08

8

8

lc

(

(8lc

U R0≦b

• Hlzfychzk}f wzhzfbfcz om 9#>99

• Jfyk}fj hio}f kc Wfgz# 9#79, vcjfy

Moybvlh 9#77#6,

• $ +$ +

$ +$ +

⎦⎭

⎪⎩⎧

⎨≠≠2

%2

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙2∥ ∘ %

∙∙∙

2

0,08,9,###,8

(8

(88

8

88

0

hcc

b

h

t

e c

fjtf t

e

e c

e

e ht

htb

c

c

η

η

η η η

COZF1 vclkef 9#>99, km ,8

,0

8

%22

%

c t K b

c

h

zyk}khl yfwvlz#

Wfgz 9#>98 oy cf~ Wfgzkoc

$ + $ + +$lklclclclc tc t tjt tcc ∙2∥


tff t

jtjw tw

tw 2222 lc,,lc # Zdfc,

$ +∥ ∥ 22 jwwcfw t K wc lclc

i{ yflhzkoc tc tc lonlon 2

• Kczfnyhzkoc i{ phyzw1

$ +

$ +

$ + $ + $ + $ +

$ + $ + #4#9:0#8i{lklclc

lcFklclcFklclc

,lclclc,,,lccnWviwzkzvzk

lc

,

,lc

tc t t

tc t twc t t K

twc tf t

jtjw tw

jww

fcwcf}jvv}

fjwf}jwfj}

jww

cjvwcv

c

cc

cw

w

w

www

∙

2∙2∙2

2222

∙2∙

222

22

∥ ∥

∥

]fykmkfj Bhzdfbhzkgh

=7

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

Wfgz 9#>98 oy cf~ wfgzkoc

$ +$ +

8

+$lklclclclc

88

%

∙2

%%

∥ c

tb t tjt t t

cbcbc

Moy 8∙2c

$ +$ +U Rb t tjt

t

t b

b

∙2∥ lclclclclc

• Bhzd#gob ghlgvlhzkocw5 whbf

hppyohgd hw moy hio}f kczfnyhl,

$ + $ + +$lklclclclc tc t tjt tcc ∙2∥

• Moy mkywz kczfnyhl, gdhcnf om }hykhilf1

$ +

$ +

$ + $ +

$ +∥ ∥ ∥

%%%

%%

222

22222

jwwbfjwwfjww t K

f t tff t

jtjw tw

wcbwcbc

wcc tw

lclclc

,,,lc

888

88lc

i{ yflhzkoc tc tc lonlon 2

• Zdfc, kcz# i{ phyzw ~kzd,$ + #,lc 8

jwfj}wbvwc%22

COZF1 moy ,,0 cbc 22 wff hio}f

kczfnyhl,

$ + $ + +$lklclclclc tc t tjt tcc

∙2∥

• Moy 18∙2c gdhcnf om }hykhilf -

kcz# i{ phyzw5

$ +

$ + $ + tb t t t

jtb t t

jw

w

wbwbw}jwv}

wjw}jwj}

jww

bw,jvbv

wjwbjw t

w t K

t

jtjw tw

bb

b

lclclclclclclc

lc

,

lc

1phyzwi{kcz#

#lc+lc$

,lc

∙2∙

2∙2∙

222

22

22

22

∥

∥ ∥

∥

∥ ∥

=:

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

Wfgzkoc 9#>9:

$ +

$ +$ +

$ +$ +

$ +$ +

$ +

$ +$ + $ +

$ + $ +$ + $ +∥

∥

∘

∥

%%

%%

%∙

∙∙

∙

2

%

%%⎦

⎭

⎪⎩⎧

⎨%∙2

%%∙2

∙

%%⎦

⎭

⎪⎩⎧

⎨ %

∙

∙∙∙

2

8

9

9

8

7

88

9

8

8

8

9

0

8

lk9

8

lc9

8

lc9

8

lc

lk8lclc

lk(8

8

lc

8

(8

(9

lc

cc

c

c

cc

c

b

e b

e b

e

c

b

c

tc

t t t

t

jt t

tc t

t

t

jt t

tb

c

t

c

b

e b t

t

jt t


$ + ,,,,lc 88 wccwf t tf

t

jtjw tw

%% 2222 zo nk}f1

$ +

∥ ∥ %

22 jww

f

w

tjw t K

b

wc

b

c 8

• Kc hio}f kczfnyhl, Wfgz# 9#79,

$ + $ + $ +

888lcwfz

,(

Fk(

(88

0

∙2%2222

%⎦⎭⎪⎩

⎧⎨ ∙∙∙2∥ ∘

∙

2∙

b ,qc ,h t,c tw

qchth

t

h qe q

cfjt

tf

c

q q

e

e qc

e ht

b

htcc

hcj wviwzkzvzf moy wolvzkoc om $ +

∥ %

2 jww

f K

b

wc 8

#

• Hlzfychzk}f wzhzfbfcz om wolvzkoc1

$ +$ +

$ +$ + $ +$ +

$ +$ + $ +

8

8

9

08

9

8

lk(8

8

lc898

8

lc

%∙

∙

2%

∙∙%

∙

%

%

⎦⎭

⎪⎩⎧

⎨

%∙∙

%∙2 ∘∥

c

b

b

e

e

e b

c

b

c

tb

c

te bb

c t

t

jt t

E

ihwfj oc goc}fyzkcn$ +

∥ %

2 jww

f K

b

wc 8

kczo nhbbh

mvcgzkoc }kh Moybvlh 9#796#4, ∥ ,jt t

fb

htc

zo nk}f, $ + $ + $ +U R tcbcbb

lc8,8888 ∙∙%∙Η%∙

∙, hcj

vwkcn Moybvlh 4#769 6, $ + tc ,8%∙Η , ~dkgd

pyojvgfw h jkmmfyfcz oyjfykcn om zdf wfykfw zfybw#

COZF1 9#>9:#9 mollo~w km b 2 8#COZF1 17,92b wviwzkzvzf kczo ftphcwkoc moybvlh

Moybvlh 9#>9:#9

Gdhcnf Moybvlh 9#>9:#9 zo yfhj1

$ +U R $ +8lklc8Fklc

%

∥ 2%2 c

c

t tc t

jt t

Wfz #,8

,lc lc tff

tjt

jw tw

tw 2222

Zdfc,

$ +$ +U R

$ +U R $ +$ + $ +,lklklc8Fk

8Fk

8lc8

8

%%

%

22%2

%222 ∥ ∥ c tc

cwc

tf tc

wcjww

fjw

w

tt K

i{ Moybvlh 9#796 >,

$ +,Fkc

htjt

t

fcht

c

2∥ Moybvlh 4#9:0#8, $ + $ + tf t lkFk 2 ,

hcj yflhzkoc #lc th thf 2

COZF1 cf~ }fywkoc yflhzfw ftpocfczkhl" hcj

lonhykzdb"kczfnyhlw, $ + $ +,lk-Fk t t Moybvlh 4#988#8#

=6

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

Cf~ Wfgzkoc

$ + $ +

$ +

$ + $ +

$ + $ +$ + ⎦

⎦⎦

⎭

⎪

⎩⎩⎩

⎧

⎨

%

%%

∙%%

%

2%%

∥

∥

∙%

%

jt tg t

tgithc

tgith

ib

jt tgith

cb

cb

cb

lc

lclc

lclc

8

8

lclc

88

8

Moy 8∙2b $ + $ + $ +

$ + $ +$ +∥

∥

%

%%

∙%%

2%

%

∙

jt tg t

tgith

i

c

i

tgithjt

ith

tg

c

cc

lc

lclclc

lclclclclc

8


$ +$ +

$ +$ +

$ +

∥

∥

2∙

%

%22%2

%

%⎟ ⎠

⎓⎜⎒

⎕

2%2

%%2

∙

}jvv}

bi

ithjww

i}jtithj}

jt tg

tg t

c

jv tgv

b

bithw

b

c

c

8

8,

lc

lclc8

,lclc

8

8

$ +

$ + $ +

$ + $ +$ + ⎦

⎦⎦

⎭

⎪

⎩⎩⎩

⎧

⎨

%

%%

∙%%

% ∥ ∙%

%

jt tg t

tgithc

tgith

ibcb

cb

lc

lclc

lclc

8

888

8

• Moy 18∙2b

Kczfnyhzkoc i{ phyzw ~kzd,

$ + #,lclcith

jtj} tgv

c

%2%2

COZF1 h jkmmkgvlz $zo yfjvgf mvyzdfy+ ivz nfcfyhl

kczfnyhl vwfmvl moy jfyk}kcn ftkwzkcn moybw i{ wfzzkcncbgih ,,,, hw jfwkyfj#

Cf~ Wfgzkoc

$ + $ +

$ +

$ + $ +

$ + $ +⎦⎦⎦

⎭

⎪

⎩⎩⎩

⎧

⎨

%

%%

∙%%

%

2%%

∥

∥

∙%

%

jtetg

etgithce

etgith

ib

jtetgith

cb

cb

cb

88

8

lc

lc

8

8

lc

Moy 8∙2

b $ +

$ + $ +

$ + $ +⎦⎦⎦

⎭

⎪

⎩⎩⎩

⎧

⎨

%

%%

∙%%

2%

%

∥

∥

∙

jtetg

etgithce

etgith

i

jtith

etg

c

c

c

8

lclc

lclc8

lc


$ +$ +

$ + $ +

$ +$ +

$ +$ + $ +

$ +$ + $ +

∥

∥

∥ ∥

%

%%

%∙

%%%

2

∙

%

%

22%2%2

%

%2%2

∙%

%

%

%2

∙

jtetg

etgith

bi

ce

etgithib

}jvv}

bi

ith

jwwi

jtith}jtithj}

jtetg

etgce jvetgv

cb

cb

b

bithw

bb

c

c

88

8

8

8

lc

8

lc8

8

8

8,

lc,lc

• COZF1 yfjvgfw zo 9#9>4#8 km

#8,,,,8,0 222222 cie hgbbih

• Zfwz1 km

∥ ∙22222222 #lclc,8,0 t t t tjt K e icbgh

• Moy 18∙2b

Kczfnyhzkoc i{ phyzw1ith

jtj}etgv

%2%2 +,lc$

• Yfjvgfw zo 9#>98#9 km #8,0 2222 e ihg

• COZF1 h jkmmkgvlz $zo yfjvgf mvyzdfy+ ivz nfcfyhlkczfnyhl vwfmvl moy jfyk}kcn ftkwzkcn moybw $wvgd

hw 9#>88, 9,>99, 9#>77, 9#>9=+ i{ wfzzkcn

cbe gih ,,,,, hw jfwkyfj#

==

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

Wfgz# 9#>9> oy Cf~ Wfgzkoc

$ +$ +

$ +

$ +$ +

$ +

$ + ⎦

⎦⎦⎦⎦

⎭

⎪

⎩

⎩⎩⎩⎩

⎧

⎨

%

%%

∙

∙

2%

∥

∥

∙

∙

∙

jtith t

tc

ith

t

bi

jtith

t

b

c

b

c

b

c

8

8

8

lc

lc

8

8

lc

Moy 82b

$ +

$ + $ + $ + $ +∥

∥ %

∙%

2%

∙

jt t

ith t

i

c

i

ith t

ith

jt t

cc

c

lclclclc

lc

8


$ + $ + jtithj} tvbc ∙

%22 ,lc

• COZF1 9#>9>#8 yfwvlzw km c 2 8#

• COZF1 bh{ if jfyk}fj myob$ + $ +

$ +$ + $ +

$ + $ +∥ ⎦⎭

⎪⎩⎧

⎨∥

%

%%∙%%

%

2%%

∙%%

jtetg

etgithce etgith

ib

jtetgith

cb

cb

cb

888 lc

lc8

8

lc

km #,8,0 bbe g ∙222

• b 281

Kczfnyhzkoc i{ phyzw ~kzd $ + #,lcith

jtj} tv

c

%22

Cf~ Wfgzkoc

$ + $ + $ +∥ ⎦⎭⎪⎩

⎧⎨∙2 ∥ %

∙

∙∙ jt t

tc t

tb

c

b

c

b

c

bjt

t t

8

8lclc

88lc

Moy b 2 8, wff 9#>98#9

$ +$ +

$ +$ +

⎦⎭

⎪⎩⎧

⎨

∙∙∙2 ∘∥

2%

∙

∙

c

e

e

e c

bb

c

b

t

e c

c

tjt

t

t

0

888

lc

(

(8lc

$ +∥ ⎦⎭

⎪⎩⎧

⎨

∙%

∙∙2

∙ 988

8

8

lc8lc

bb

t

tjt

t

tbb

$ +

$ +$ + $ +

⎦⎭

⎪⎩⎧

⎨

∙%

∙%

∙∙

2

∙

∥

79

9

8

9

8

9

8

lc9

8

lc8

lc

bb

t

b

t

t

jt t

t

b

b

$ +

$ + $ +

$ + $ +

$ +⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩

⎧

⎨

∙%

∙%

∙%

∙∙

2

∙

∥

:

79

97

8

7

8

=

8

lc=

8

lc7

8

lc

8

lc

b

b

t

b

t

b

t

t

jt t

t

b

b

Moy b 2 8 hcj 8∙2c , wff 9#>98#7

• Mkywz kczfnyhl1

Wfz h 2 0, i 2 8 kc hio}f kczfnyhl,

$ +$ + $ +

$ +$ +

$ +$ +∥ ∥ ⎦

⎭

⎪⎩⎧

⎨

%%

%∙

∙2

% ∙

∙

∙jt

ith t

tc

ith

t

bijt

ith

tb

c

b

c

b

c

8

8

8

lclc

8

8lc

• Moy 182b wff 9#>98#9 ~kzd #cb ↙

• Ftphcwkoc moybvlh1 Gdhcnf om }hykhilf5

$ +

$ +∥ ∥

∙

∙∙

22

2222

#

,,,lc

8

88

wbc

b

c

wbbw

fw tjw t

w K

f t tf t

jtjw tw

Co~ vwf kcjfmkckzf kczfnyhl pyf}kovwl{jfyk}fj $obkzzfj kc NY >, ivz wvibkzzfj hw

fyyhzvb+,

$ + $ +c

b

ch

htjtf t

c

htbc 8

,,8

8 %2

∙Η∙2∥

∙

η η η

η

~kzd phyhbfzfyw U R8,8 %2∙2 cbh η zo nk}f1

$ +$ +

$ +U R

$ +U R

$ +$ +

$ +$ +U R

$ +$ +U R

$ +#8

4#769#9,i{(

lc88

8

(

(

lc8

8

(8

lc8,8

lc8,88

8

088

08

lc8

8

8

,tc

e

tb

tb

b

e

tb

b

fb

b

tbc

tbcb

K

c

e

e

bc

c

e

e

c

tb

c

c

c

%Η

⎦⎭

⎪⎩⎧

⎨ ∙

∙∙2

⎦⎭

⎪⎩⎧

⎨ ∙

∙∙2

∙

∙%Η∙2

∙∙%Η∙

∙2

∘

∘

2∙%

2%

∙∙

%

%

∙

^ykzkcn wvbbhzkoc kc yf}fywf oyjfy1

=>

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

$ + $ +$ +

$ +$ +$ + $ +∥ ⎦

⎭

⎪⎩⎧

⎨

∙%%

∙

∙%

∙%

∙∙2

%

∙∙

∙ 87

9

9

8

88

(

8

lc8

8

lc

8

lc8lcc

ccc

bb

c

b

c

b

tcc

b

tc

b

t

tjt

t

tE

$ +$ +$ +

⎦⎭

⎪⎩⎧

⎨

∙∙∙2 ∘

2%

∙

∙

c

e

e

e c

bb

t

e c

c

t 088

8

lc

(

(8i{ kcwpfgzkoc#

COZF1 zdkw moybvlh bh{ if jfyk}fj yfhjkl{ i{yfwzhzkcn

$ + $ +jwfwjt

t

t K

wbc

b

c

∥ ∥ ∙∙22 8lc

hcj hppl{kcn kczfnyhl

jfyk}fj hio}f kc Wfgz# 9#79,

$ +$ +

$ +⎦⎭

⎪⎩⎧

⎨2

%2

⎦⎦⎭

⎪

⎩⎩⎧

⎨

∙∙

∙∙2∥ ∘

∙

2%

∙∙∙∙

8,9,###8

(8

(88

08

8

c

b t

e c

fjtf t

e

e

e c t

tb

c

c

η ι η

η η η ι ι

~kzd phyhbfzfyw$ + #lc,,8,,8 8

t twf tccbcwbc 222%2↙2 ∙∙ ι η

• Moy 17,9,82c wviwzkzvzf kczo ftphcwkoc moybvlh

hio}f

• Moy b 2 8 hcj 8∙2c , wff 9#>98#7#

Wfgz# 9#>94

$ +

$ +

$ +⎦⎦⎦

⎭

⎪

⎩⎩⎩

⎧

⎨

%

%

∙%

%

2%

∥

∥

∙%

%

jtith

ith tci

ith t

b

jtith t

cb

cb

cb

88

8

lc

lc

8

8

lc

Moy 8∙2c

$ +

$ +∥

%2

% i

ith

ith

jt lk

lc

Moy 8,8 ∙22 bc , wff 9#>94#9 # Hlwo, wff

Wfgz# 9#79, hio}f, moy bvlzkplf wolvzkocw zo

$ +∥

%jt

t

ithlc$kc zfybw om jklonhykzdbw,

jfmkcfj kc Bhzd# Wvbbhy{+ hll om ~dkgd

jkmmfy myob fxvk}hlfcz NY wolvzkoc(

• Cf~ yfjvgzkoc moybvlh5 whbf yfjvgzkochppyohgd hw Moybvlh 9#>94#8


$ +$ +

$ +$ +

∥

∥

∥

%

%

%∙%%2

∙

%222

%

%2%2

∙%%

%

∙

jtith

ith t

b

ci

ithb

t

}jvv}

b

tjt t}jt tj}

jtith

ithcijvithv

cb

c

b

b

bb

c

c

888

8

8

lc

8lc8

58

,

,lc

,lc

• 18∙2c

o Gdhcnf om }hykhilf1

$ + $ + $ +i

ith

i

v

i

v

v

jv

i K

ijtjvithv

%2222

2%2

∥ lklklcFk

lc

8

,

i{ Moybvlh 4#9:0#8#

]fykmkfj bhzd#gob

$ +$ +U R $ +U R

$ +U R $ +9

9

9

lklk

lcFk9lcFk

lc

i

ithhith

i

ithhith

jtith

t

%∙%

2%∙%

2%∥


$ +⎦⎭

⎪⎩⎧

⎨∙2

∙2

∙

22%2

∥ ∥ ∥ w

jwhjw

w

w

ijw

w

ihw

i K

i

hw

tijtjwithw

lclc

8

lc

/ 8

,,

9

o jww

w∥ lc

1

=4

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

$ +

$ + $ +U R $ +U R9

99

lklc9Fklc9Fk

9Fk

,8

,lc

ithithw

vjwv

fjv

v

w K

wfwjw

jvwv

v

v

%2%2

2222

222

∥ ∥

i{ 4#988#8 hcj yflhzkoc #lc th thf 2

o ∥ w

jw

h lc 1

$ +

$ + $ +U R $ +ithhithhwh

vhjvv

fhjw

v

wh K

wfwjw

jvwv

v

v

%2%2

2222

222

∥ ∥ lklcFklcFk

Fk

,8

,lc

i{ 4#988#8 hcj yflhzkoc #lc th thf 2

• $ +

$ +U R $ +U R∥

%∙%2

% 9

lcFkith9lcFk

lc i

ithh

ith

tjt

COZF1 km

$ + 9#9#>9:i{lklc

,8,0 9 tjt

t

t K ih 2222 ∥

}fykmkfj bhzd#gob

Cf~ Wfgzkoc

$ +

$ +$ +

$ +$ +

jt th t

th

c

b

tc

th

jt t

th

c

b

c

b

c

b

∥

∥

%

%

∙%

∙

%∙

2%

∙

∙ lc

lclc

88

lclc

lclc

8

8

Moy 82c

$ +

$ + $ +

$ +∥

∥

%∙

%%

2%

∙

jt t

thb

th th

jt t

th

b

b

b

lclc

lclclc

lclc

8

Moy 8hcj8 22 cb

Wff kczfnyhl iflo~,$ +

$ + $ + t th th

jt t

th

lclclclc

lclc

∙%%

2%

∥


$ +

$ +hc

wc

c

hwh thw

f

f t

ff t tfff

t

jtjw thw

∙

∙∙

∙%

2

2⇙22

2%2

8

88

lc

,lc

$ +$ +

$ +$ + $ + $ +∥ ∥

∥ ∥ ∙2

∙

∙

∙

2

222

#lc8

lc8

lclc

88

8

8

jwwff

jwwff

jww t t

jww t K

bew

eh

ce bwc

hc

bc

c

b

• Kczfnyhzkoc i{ phyzw oc $ +∥ 1lc jwwfbew

$ +$ +

$ + $ +∥

∥ ∙

∙

∙2∙

222

22

w

jwwf

e

b

e

wf}jvv}

fe

jwf}jwfj}

jww

wbjvwv

bewbew

ewewew

b

b

8

8

lclc

8,

lc,lc

Wviwzkzvzkcn

=;

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

$ + $ +

$ +

$ + $ +

$ + $ +$ +

$ +$ + $ +$ +#

lclclc

88lclc

lc

lclc

8

8

lclc

8

,,8,,lc

8

8

888

88

8

888

∥

∥

%%∙%∙ %∙

2

⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩

⎧

⎨

%

%

∙∙

∙

%

2

2∙22%2

∙

∙

∙∙∙

∙∙

∙

∙∙∙

jt th t

thc

b tc

th

jt th t

thf t

c

b

c

thf t

f K

f tfce t

jtjw thw

c

b

c

b

bhcc

bhcc

hc

hccwc

COZF1 fhwkfy hppyohgd1 kcz# i{ phyzw oc

$ +c

b

t

jtj} thv 2%2 ,lclc

• Moy 182c

$ +

$ + $ +$ +

∥

∥ %

∙%%

2%

∙

jt

t

thb th th

jt t

th

b

b

b

lclclclclc

lclc

8

jfyk}fj i{ gdhcnf om }hykhilf ~kzd

$ +

$ + #,lc

1~kzdphyzwi{ckczfnyhzkohcj,lc

nk}f,zo,,lc

jwj}wv

jww K

t

jtjw thw

b

b

22

2

2%2

∥

Bhzdfbhzkgh pyo}kjfj wolvzkoc kc zfybw om

d{pfynfobfzykg mvcgzkoc#

Cf~ Wfgzkoc

$ +

$ +

$ +⎦⎦⎦

⎭

⎪

⎩⎩⎩

⎧

⎨

%

%∙

%

%

2%

∥

∥

∙

%

jt th

th tb

th t

c

jt th t

bc

bc

bc

lc

lclc

lclc

8

8

lclc

8

8

Moy b 2 8

$ +

$ +$ + $ +$ +U R⎡⎢

⎡⎬⎥

⎡⎫

⎡⎤⎮

%%

∙%

%

2%

%∙

%

∥

thcf

th t

c

jt th t

hc

c

c

lc8Fk

lclc

8

8

lclc

8

8

Moy 8hcj8 ∙22 cb

$ +

$ + $ + t th th

jt t

th

lclclclc

lclc

∙%%

2%

∥

• Wkbklhy zo $ +jt

t

th

c

b

∥

% lclc hio}f1 bhef #cc ∙↙

• Moy b 281


$ +

$ +

∥ ∥ 22

22

2⇙22

2%2

%

%2

%

%%

∙%

jwwff

w t K

f

f

f

f t

ff t tfff

t

jtjw thw

ew

eh

c

eh

ewce

hc

wc

c

hwh thw

lc8

lc

,lc

8

8

8

88

lc

o Kczfnyhzkoc i{ phyzw1

>0

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

$ +#

Fklc

8lc

8,

,lc

e

ew

e

cwf

jww

f

e e

cwf}jvv}

fe

}jwfj}

w

jwjvwv

ew

ewew

ewew

∙

2∙2∙

22

22

∥

Wviwzkzvzkcn

$ + $ +

$ +

$ + $ +

$ +$ +

$ + $ + $ +

$ +U Rhcchcc

hcc

hc

hccwc

f tf th tc

thcc

c

thf t

f K

f tfce t

jtjw thw

8888

88

8

888

lklclc8

8

lc8Fk8

8

8

lclc

8

,,8,,lc

%%%∙%

%%

%

%%%

∙%%

2

⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩

⎧

⎨

%%%

∙%

%

2

2%22%2

i{ Moybvlh 4#988#8, $ + $ +,lkFkt

f t 2 hcj yflhzkoc,

#lc th thf 2

• Moy 18hcj8 ∙22 cb bhzd#gob ghlgvlhzkoc5

$gdhcnf om }hykhilf,+lcocphyzwi{kcz#hcjlc jww,j}v thw 22%2

Cf~ Wfgzkoc

$ +

$ +$ + $ +$ +U R⎡⎢⎡⎬⎥

⎡⎫⎡⎤⎮

∙%

∙∙

%

2∙

%∙

%

∥

h tcf

th t

c

jt th t

hc

c

c

lc8Fklclc

88

lclc

8

8

Moy 8∙2c

$ +$ + $ + t thh tjt

t

thlclclclc

lclc∙∙∙2

∙∥

• Wkbklhy zo $ +jt th tc

∥ % lclc , hio}f

• 8∙2c 1

Wkbklhy zo$ +

jt t

th∥

% lclc, hio}f

Cf~ Wfgzkoc

$ +

$ +$ + $ +∥

∥

∙∙ ∙∙∙

∙∙

2∙

88 lc8

8

lc8

lc

cc

c

th

jt

c thc

t

th

jt

Moy 82c

$ +∥ ∙∙2∙

h tf th

jt h lcFklc


∥ ∥ ∙

∙∙∙

∙

∙2∙2

22

222

∙2∙2

jww

ffjw

w

t K

tffff t

t

fffff

t

jtjw thw

c

w

h

c

hwwh

h

th thw

,

,lc

8lc

lc


>8

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

$ +

$ + $ +$ +

$ + $ +

$ +$ +

$ +h tc

f

thc

e c t

th

jt

hc

c

e

e c

e c

c

∙∙

∙

%∙∙

∙∙∙∙

2∙

∘

∥ ∙

2∙∙

lcFk(8

8

lc(8

(988

lc

9

08

$ +$ +h tf

th

t

th

jt h ∙%∙

2∙∥ lcFk

lclc9

$ +

$ + $ +

$ + $ +

$ +$ + $ +∥

∥

∥

∥

∥

∙

∙

∙

∙

∙2

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙∙∙

∙∙

∙

∙

∙∙

∙∙∙

2∙

∙∙

∙2

∙∙

∙∙

∙∙

2∙

∙∙2

∙222

∙22

#lc8

8

lc8

88

88

8

8

8,

,

88

lc

88

88

8

8

jt th t

tf

c thc

tf

jwwc

f

wc

f

jwwc

f

wc

f

}jvv}

wcc

wjww}jwwj}

jwfjvfv

c

h

c

h

thw

c

w

c

w

c

w

c

w

c

c

cc

ww

$ +$ +

$ +

$ +$ + $ +∥

∥

∙∙

∙

∙

∙

∙

∙∙∙

∙∙2

⎦⎦⎦⎦⎦

⎭

⎪

⎩⎩⎩⎩⎩

⎧

⎨

∙

∙

∙∙

∙∙∙

∙2

88

8

8

lc8

8

lc8

lc8

8

lc8

cc

c

h

c

h

h

th

jt

c thc

t

jt th t

tf

c

thc

tf

f K

• Moy 182c Gdhcnf om }hykhilf5

$ + $ +h tfwfjww

ff K

ff t

t

fffff

t

jtjw thw

hh

w

h

wh

h

th thw

∙∙2∙∙2∙2

2

⇙222∙2∙2

∥ ∙

∙

∙

lcFkFk

#

,,lc

8lc

lc

i{ 4#988#8 hcj yflhzkoc, #lc th thf 2

• Kczfnfy"ftphcwkoc moybvlh1Myob hio}f,

∥ ∙

∙2 jww

ff K

c

w

h

Hppl{ kczfnyhl moybvlh, hio}f, Wfgz# 9#79,

$ + $ + $ +$ +

$ + $ + ,###7,9 ,###,9,8

8

(

Fk8

(

(88

8 8

0

⎦⎭

⎪⎩⎧

⎨222

∙∙%

⎦⎭

⎪⎩⎧

⎨ ∙∙∙

∙2∥ ∘

∙

2∙

∙∙

bc

b" q

qc

t

t q

e q

c

fjt

t

f

c

q q

q

e

e qc

e e

t q

b

tcc

ι ι

ι ι ι

~kzd phyhbfzfyw,

U R8,8,,8,lc ∙2222∙22 c qcbc th tw ι

o Hlzfychzk}f ftpyfwwkoc om ftphcwkoc

moybvlh1

>9

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

$ +

$ +$ +

$ +$ + $ +$ +

$ +$ +$ +(8

lcFk8

lc898

88

lc

9

08

∙

∙∙

%∙%∙∙

∙∙

2∙

∘

∥ ∙

2%

c

h tf

the cc t

th

jt

hc

c

e

e

e

c

c

E

ihwfj oc ftphcwkoc moybvlh moy ∥ ∙

jt tf

b

tc ι

pyf}kovwl{

wvibkzzfj hw Moyb# 9#796#8;, ~dkgd pyojvgfw h

jkmmfyfcz oyjfykcn om zdf wfykfw zfybw#

• 92c 1 Kc hio}f kczfnyhl,

$ + $ +$ + $ +∥ ∥ ∙∙ ∙∙∙

∙∙2

∙ 88lc8

8

lc8lcccc

th

jt

c thc

t

th

jt

,

wfz ,92c hcj cozf,

$ +$ +h tf

wfjww

ff

th

jt

h

h

w

h thw

∙∙

2∙∙2∙2∙∥ ∥

∙∙2

lcFk

Fklc

lc

COZF1 Moybvlh :#989#: yfwvlzw moy zdkw kczfnyhl~kzd lkbkzw myob 0 zo 8#

Cf~ Wfgzkoc

$ +

$ + $ + $ +∥

∥

∙∙∙ ∙

∙∙

∙∙

2

888 lc8

8

lc8

8

lc

bcbc

bc

t t

jt

b

c

t tb

t t

jt

Moy 82b

8lk

lc 8 ⎟ ⎠

⎓⎜⎒

⎕ 2∥ ∙

t t t

jtcc

$ + $ +$ +∥ ∙∙

∙2

2

8lc8

8

lc

8Moy

bb tb t t

jt

c

#79#>98wff,8Moy22 bc

• Mkywz kczfnyhl1


$ +wccwf t tf

t

jtjw tw

∙∙ 2222 88,,,lc zo nk}f,

$ + $ +

∥ ∥ ∥ ∙∙∙

222 jww

fjw

w

fjw

w t

t K

b

wc

b

wc

bc

88

o Kczfnyhzkoc i{ phyzw1

o

$ + $ + $ +

$ +$ +

$ +$ +$ +

$ +

$ +$ +

$ +$ + $ +

$ + $ + $ +$ + $ +∥

∥

∥ ∥

∥

∙∙∙

∙

∙

∙

∙

∙

∙

∙

∙

∙

∙∙

∙∙

∙

∙∙∙∙2

∙

∙∙

∙∙2

∙∙

∙∙∙

∙∙2∙

∙∙222

∙22

888

8

8

8

8

8

8

8

8

8

88

lc8

8

lc8

8

lc8

8

lc8

8

8

8

8

8,

8,

bcbc

b

c

b

c

b

wc

b

wc

b

bb

wcwc

t t

jt

b

c

t tb

t

jt

t

t

b

c

tb

t

jww

f

b

c

wb

f}jvv}

wbjww}jwwj}

fcjvfv

COZF1 wfzzkcn , ccbb ∙↙∙↙ kc 9#>98#8

pyojvgfw h pooy yfjvgzkoc moybvlh wkcgf po~fyw om

tlc hyf kcgyfhwkcn# Wfzzkcn cc ∙↙ kc 9#>9:

pyojvgfw jfwkyfj yfwvlzw#

• Moy b 2 81

>7

7/23/2019 500 Integrals


‚Wfgz# 9#>8 - 9#>9"9#>7‚

$ +

$ +$ +

$ +$ +

$ +

$ +$ +$ +

⎟ ⎠

⎓⎜⎒

⎕

∙

∙∙

%⎦⎭

⎪⎩⎧

⎨ ∙

∙

∙∙∙∙

2

∙

∙∙

∙∙

∙

2∙ ∘

∥

8

8lk

(8

88

lc

8

(8

(98

8

lc

88

8

9

08

c tb

c

t

c

b

e b

t

t t

jt

b

b

e b

e b

e

e

c

bc

U R83c

$ + ,,,,lc 88 wccwf t tf

t

jtjw tw

∙∙ 2222 zo nk}f

$ +

$ +U R $ +U R

U R ⎦⎭

⎪⎩⎧

⎨22

2∙222

∙

∙

∙∙∙

∥ ∥

8

8

lc888

8lklk

lklc8Fk

c

c

tc

wcc

t t

f tcjww

fjw

w

t K

o COZF1 kz kw vc~kwf zo wfz c 2 8 kc zdkwmoybvlh zo wol}f Moybvlh 9#>98#7,

∥ t t

jt

lc

82

2c

lk$8+#

• Moy 182c

$ + $ +$ + 8lc8

8

lc∙

∙∙2∥ bb

tbjt

t t

jti{ gdhcnf om

}hykhilf,#lc tw 2

• Kczfnfy"ftphcwkoc moybvlh1

Jfmkcf myob hio}f,$ +

∥ ∙∙

2 jww

f K

b

wc 8

hcj vwf kczfnyhl, ,jt t

fb

tc

∥ ∙ ι

Wfgzkoc 9#79, hio}f,

$ +$ +

$ +

$ + $ + ,###7,9,###,9,8

8

(

Fk8

(

(88

8

0

⎦⎭

⎪⎩⎧

⎨22

∙2

∙∙

%⎦⎭

⎪⎩⎧

⎨ ∙∙∙∙2∥ ∘

∙

2∙

∙∙

bc

b q

qc

t

t q

e q

c

fjt

t

f

c

q q

q

e

e qc

e e

t

b

tcc

ι ι

ι ι ι

~kzd phyhbfzfywU R#8,8,8,lc ∙2∙2222 b qcc tw t ι

• Hlzfychzk}f moyb moy ftphcwkoc1

$ +2∥ bc

t t

jt

lc

$ + $ +$ +$ + $ +$ +

$ +$ +$ +

⎟ ⎠

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7/23/2019 500 Integrals


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7/23/2019 500 Integrals


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7/23/2019 500 Integrals


YFMFYFCGFW

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9>6

7/23/2019 500 Integrals


Boll, ]kgzoy D#, Ahwoc Yowfcifyn, Hybkc Wzyhvi, Phz ^dkz~oyzd# Zdf Kczfnyhlw kc Nyhjwdzf{c hcj

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9>=

7/23/2019 500 Integrals


KCJFT OM W[BIOLW, MVCGZKOCW,

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7/23/2019 500 Integrals


H

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9>4

7/23/2019 500 Integrals


F

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9>;

7/23/2019 500 Integrals


kcgobplfzf $ +$ + t∞

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940

7/23/2019 500 Integrals


bhzdfbhzkghl wvbbhy{ 8"98Bfkafy‛w N Mvcgzkoc 9=:

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948

7/23/2019 500 Integrals


wkc wkcf 8>:, 979"974, 968wpfgkhl mvcgzkocw 8"98, 909"9>6

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949

7/23/2019 500 Integrals


KCJFT OM MOYBVLHW

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947

7/23/2019 500 Integrals


∥ jt t

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94:

7/23/2019 500 Integrals


∥ ∙

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946

7/23/2019 500 Integrals


∥ %

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94=

7/23/2019 500 Integrals


$ +∥ %

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94>

7/23/2019 500 Integrals


∥ ∙ th

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944

7/23/2019 500 Integrals


jtht

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94;

7/23/2019 500 Integrals


$ + $ +∥ ∓

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9;0

7/23/2019 500 Integrals


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9;8

7/23/2019 500 Integrals


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9;9

7/23/2019 500 Integrals


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7/23/2019 500 Integrals


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∙∙

0

89 9

jtf t tc ψ

7#:=8

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0

9

99

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fc

t ψ

7#:=8

∥ ∓

∓∙

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7#:=8

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v

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7#:=9

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t

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7#:=9

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7#:=9

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v

t jtfi tic

ι

7#:=9

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v

hb

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0

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fhb

tic

ι

7#:=9

∥ ∓

¸

∙ ¸

v

hb

t

jt t

fic

ι

7#:=9

9;:

7/23/2019 500 Integrals


∥ ¸

∙ ¸v

hb

t

jt t

fic

0

ι

7#:=9

$ +∥ ∙¸

}

v

tbjtfiht ψ

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v

tbjtfhti ψ

7#:=9

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∙∙0

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7#:=9

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v

tbjtfhti

0

ψ

7#:=9

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∙∙v

tbjtfhti ψ

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∙}

v

c

t

jtiht

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7#:=9

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∙}

v

c

t

jthti

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7#:=9

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fc

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7#:=9

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c

t

jthti

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0

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7#:=9

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7#:=9

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∓∙

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7#:=9

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ptbjtfiht

0

7#:=9

$ +∥ ∓ ∙

¸0

jtiht

fc

pt

7#:=9

9;6

7/23/2019 500 Integrals


$ +∥ ∓ ∙

¸v

c

pt

jtiht

f

7#:=9

$ +∥ ¸

∙v

c

pt

jtiht

f

0

7#:=9

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i

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7#:=9

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⎟ ⎠

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7#:=9

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7#:=9

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7#:>8

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i

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∥ ⎟ ⎠

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v

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0

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∥ ⎟ ⎠

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v

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t0

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%∙}

v

cjtiht ι ftp

7#:>8

9;=

7/23/2019 500 Integrals


$ +U R∥ ∙

%∙v

cjtiht

0

ftp ι 7#:>8

$ +U R∥ ∙

%∙v

jtiht0

8ftp ι

7#:>8

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7#:>8

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v

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∥ ⎦⎭

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∥ ⎟ ⎠

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7#:>8

9;>

7/23/2019 500 Integrals


$ +∥ ⎟

⎠

⎓⎜⎒

⎕ ∙

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∙}

v

jt t t

th ι μ

μ

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8

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lc

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9;4

7/23/2019 500 Integrals


∥ ∙

8

0lc

lc

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tjt

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vth

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9;;

7/23/2019 500 Integrals


jt

t

tc

p

∥ ⎟ ⎠

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∙8

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8

8lc

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$ + lc

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700

7/23/2019 500 Integrals


$ +∥ ⎦⎭

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8

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708

7/23/2019 500 Integrals


$ + thlk 4#9:8

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709

7/23/2019 500 Integrals


⎟ ⎠

⎓⎜⎒

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9

8

4#778

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$ + $ + t t ∙Η%Η 88 4#77:

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4#77;

707

7/23/2019 500 Integrals


⎟ ⎠

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9

c

4#77;

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,:

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4#769

$ + $ + $ + $ + t t t t ,:,,7,,9,,8 η η η η

4#769

$ + $ + $ + $ + t t t t ∙∙∙∙ ,:,,7,,9,,8 η η η η 4#769

$ + t,θ η 4#767 KCZFNYHL YFPYFWFCZHZKOCW $9=8"9=9+

$ + { t ¸,θ η 4#767

$ + t{,θ η

⎟⎟ ⎠

⎓⎜⎜⎒

⎕

{

t,θ η

4#767

70:

7/23/2019 500 Integrals


$ + t,θ Η 4#767

⎟⎟ ⎠

⎓⎜⎜⎒

⎕ Η

{

t,θ

4#767

$ + { t ¸Η ,θ 4#767

$ + te h ,¸Η 4#76= MVCGZKOCHL YFLHZKOCW $9=7"9=:+

$ + te h ,¸η 4#76=

$ +jh

thj ,Η 4#76=

$ +jh

thj ,η

4#76=

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j fym

4#76=

$ +U R $ +U R tjt

j t

jt

j lk - Fk

4#76=

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4#76=

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tj Η 4#76=

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4#76=

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j

0

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ι

ι β η

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v

jv

j

c

c,8

4#76=

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ι cv

jvj

c

c

,8Η

4#76=

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ι η

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v

j

j c,Η

4#76=

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cv

j

j ,

4#76=

⎟ ⎠

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⎠

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t t ,9

8,,

9

8η

4#76; YFLHZKOCWDKPW KZD OZDFY MVCGZKOCW

$9=6"9>9+

$ + ⎦⎭

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9

8 thη

4#76;

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9

8

4#76;

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9

8 t

4#76;

⎟ ⎠

⎓⎜⎒

⎕ t,

9

8η

4#76;

706

7/23/2019 500 Integrals


⎟ ⎠

⎓⎜⎒

⎕ ¸ 9,

9

8 tη

4#76;

⎦⎭

⎪⎩⎧

⎨⎟ ⎠

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⎠

⎓⎜⎒

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8,

9

88 t tη

π

4#76;

$ +∓¸fym 4#76;

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⎠

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c,

9

4#76;

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e

c,

4#76;

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e

c,η

4#76;

$ +∥ ∓

∙8

9

lclcjz

z

z

4#7=> ZDF PWK MVCGZKOC# FVLFY‛W GOCWZHCZ1

KCZFNYHL YFPYFWFCZHZKOCW $9>7"9>6+

∥ ∓

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lc

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8

8

z

jz

z z z

4#7=>

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lc

8

z

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z z z z

4#7=>

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z z

4#7=>

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4#7=>

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f

f

z

fz

z z

4#7=>

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fz

4#7=>

$ + $ +∥ ∓

∙∙ ∙

∙0

8Fk jz zfz

fz

4#7=>

70=

7/23/2019 500 Integrals


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500 Integrals

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