INTEGRATION of FUNCTION of ONE VARIABLE INDEFINITE ...math.bme.hu/~nagyi/calc1/integraloffofonevariable.pdfFinding the indefinite integrals Reduction to basic integrals, using the

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INTEGRATION of FUNCTION of ONE VARIABLE

INDEFINITE INTEGRAL

Finding the indefinite integrals

Reduction to basic integrals, using the rule ....)()( =⋅′∫ dxxfxf n

1. ( )dxxxx∫ +3 2. ∫

+dx

x

x

3

2)32(

3. dxx∫ +100)32(

4. dxx

∫+ 32

1 5. dx

xx∫

++ 22

1

2 6. dx

xx

x

∫++

+

22

1

2

7. dxxx

x

∫++ 22

2 8. dx

xx∫

++ 22

1

2

9. dx

x

x

∫+1

2

10. dx

x∫

−

29

1 11. ∫

+

dx

xx2

2

1 12. dx

x

xxx

∫+

2

13. ∫+

dxx

x

122

14. dxxx902 )12( +∫ 15. dxxx

243 )1( −∫

16. dxx

x∫

+

2sin

12sin

2 17. xdx∫

2sin 18. xdx∫3

sin

19. ∫ xdxtan 20. xdx∫2

tan 21. xdxex

cosh∫

22. dxxx∫2sin 23. dx

x

x

∫ln

24. dxxx

∫ln

1

25. dxxx

3/2)(ln1∫ 26. dxee

xx 10)32( +∫ 27. dxe

e

x

x

∫+ 32

28. dxx

x

21

∫ 29. dxx

x

∫+

+

)12(cos

)12sin(2

30. dx

x

x

∫−

2

2

1

)(arcsin

31. ∫ dxx)4tanh( 32. dxx)3(sinh 2

∫ 33. dx

x∫

+2

61

1

34. dx

x

x

∫−

−

21

1 35. dx

xx

xx

∫−

−+

1

12 36. ∫

+

dxxcos1

1

Right or wrong?

1. cxx

xdxx +=∫ sin2

sin

2

2. ∫ +−= cxxxdxx cossin

3. ∫ ++−= cxxxxdxx sincossin 4. cx

dxx ++

=+∫ 3

)12()12(

3

2

5. cxdxx ++=+∫32 )12()12( 6. c

xdxx +

+=+∫ 6

)12()12(

3

2

Answers:

1. c

x

x ++

45

24

2/5 2. c

xx

x +−−2

2

912ln4 3. c

x+

+

202

)32( 101

4. c

x

+

+

2

32ln

2

5. cx ++ )1arctan( 6. c

xx

+++

2

)22ln( 2

7. cx

xx

++−++

)1arctan(2

)22ln( 2

8. cxarsh ++ )1( 9. cx ++12 10. c

x

+

3arcsin 11. cxarch ++ )1(

12. c

x

x +−2

2 13. c

x

++

4

)12ln( 2

14. c

x

++

364

)12( 912

15. c

x

+−

12

)1(34

16. c

xx

+−−

2

2cot

2

2cos 17. c

xx

+−

4

2sin

2 18. c

xx ++−

3

coscos

3

19. cx +− cosln

20. cxx +−tan 21. c

xex

++

24

2

22. c

x

+−

2

cos2

23. c

x

+

3

)(ln2 2/3

24. cx +lnln

25. c

x

+

5

)(ln3 3/5

26. c

ex

++

22

)32( 11

27. c

ex

++

2

)32ln( 28. c

x

+⋅ 22ln

2

29. c

x

+

+ )12cos(2

1 30. c

x

+

3

)(arcsin 3

31. cx +)4cosh(ln4

1 32. cx

x

+

−

6

)6sinh(

2

1

33. cx

+

−

6

)6(sinh 1

34. cxx +−+2

1arcsin 35. cxx ++− ln212 36. c

x

+

2tan

Right or wrong? 1. w 2. w 3. r 4. w 5. w 6. r

Integration by parts

∫ ∫ ′−=′ dxxgxfxgxfdxxgxf )()()()()()(

1. dxexx

∫−

+ )12( 2. ∫ − dxxx )13cos( 3. ∫ arctgxdx

4. ∫ + xdxx ln)1( 5. dxx )1ln( 2+∫ 6. xdxe

x

cos∫

Answers:

1. cexx

++−−)32( 2. c

x

x

x

+−

+−

9

)13cos()13sin(

3 3. c

xxarctgx +

+−

2

)1ln( 2

4. cx

x

xx

x

+−−

+

4ln

2

22

5. carctgxxxx ++−+ 22)1ln( 2

6. cxxe

x

++

2

)cos(sin

Integration of rational functions

1. ∫+−

dxxx

x

232

3

2. dxx

x

∫+12

4

3. dxxxx

xx

∫−+

++

)2)(2(

322

4. dxxx

x

∫−

+

)2(

322

5. dxxx

x

∫+

−

)4(

122

2

6. dxxxx

xx

∫++

++

)1)(1(

53222

2

7. dxx

x

∫+1

8. dxx

x

∫+12

2

9. dxx

x

∫−1

22

3

3

Answers:

1. cxxx

x

+−+−−+ 2ln81ln32

2

2. cxx

x

++− arctan3

3

3. cxxx +−+++− 2ln8

132ln

8

9ln

4

3 4. cx

x

x +−++− 2ln4

7

2

3ln

2

7

5. cxx +++− )4ln(8

9ln

4

1 2 6. cxx

x

x +−++−− arctan31ln25

ln2

7. cxx ++− 1ln 8. cxx +− arctan 9. cxx +−+ 1ln22

Integration by substitution

1. dxx∫ −

216 2. dxxx∫ − 2

2 3. dx

xx∫

+22

1

1

4. dxxx∫ + 3 5. dxx

∫++ 121

1 6. dx

xx∫

+

1

7. dxe

e

x

x

∫+

−

1

12

2

8. dxee

e

xx

x

∫++

+

34

1

2 9. dx

x

x

∫−

4

21

10. dxx

∫+ sin1

1 11. dx

x∫

+ sin2

1 12. dx

x

x

∫− cos1

cos

Answers:

1. c

xxx

+

−+

2

41

48

4arcsin8 2. c

xar

xx +−

−−−−

2

)1cosh(1)1()1(

2

1 2

3. c

x

x

++

−

21

4. cxx ++−+2/32/5

)3(2)3(5

2

5. cxx +++−+ )121ln(12 6. cx ++ )1ln(2

7. cxex

+−+ )1ln( 2 8. c

xex

+++

−

33

)3ln( 9. cx +− )(arcsincot

3

1 3

10. c

x

+

+

−

2tan1

2 11. c

x

++

3

1)2/tan(2arctan

3

2 12. cxx +−− )2/cot(

DEFINITE INTEGRALS

Express the limits as definite integrals:

1. ∑=

→

∆

n

k

kkP

xc

1

2

0

lim , where P is a partition of [ ]2,0 .

2. k

n

k kP

x

c

∆∑=

→

10

1lim , where P is a partition of [ ]4,1 .

3. k

n

k

kP

xc ∆−∑=

→

1

2

0

4lim , where P is a partition of [ ]1,0 .

4

Answers:

1. dxx∫2

0

2 2. dxx∫4

1

1 3. dxx∫ −

1

0

24

Find the derivative of

1. ∫x

tdt

0

cos 2. dtt

x

∫ +

0

21 3. dtt

x

∫2

0

cos

4. dt

t

x

∫−

sin

02

1

1 5. dt

t

x

∫+

2

0

61

1

Answers:

1. x

x

2

cos 2. 2

1 x+ 3. xx cos2 ⋅ 4. 1 5. 12

1

2

x

x

+

Find the average value of f over the given interval. At what point or points in the given

interval does the function assume its average value?

1. 1)( 2−= xxf , [ ]3,0 2. xxf sin)( = , [ ]π2,0 3. 1)( −= xxf , [ ] [ ]3,1,1,1−

Answers:

1. 0=avef , f(1)=0 2. 0=

avef , 0)( =πf 3. [ ] 2/1,1,1 −=−

avef ,

aveff =± )2/1( ,

[ ] 1,3,1 =avef , f(2)=1

Find upper and lower bounds for the value of

1. dxx

∫+

2/1

0

21

1 2. dxx )sin(

1

0

2

∫ 3. dxx∫ +

1

0

8

4. dxx∫ +

1

0

71 5. dx

x

ex

∫+

−100

0100

6. dxx∫ +

1

0

cos1

7. Suppose that f is continuous and that ∫ =

2

1

4)( dxxf . Show that f (x)=4 at least once on [ ]2,1 .

Answers: upper bound=ub, lower bound=lb

1. up=1/2, lb=2/5 2. ub=sin1, lb=0 3. ub=3, lb= 8 4. ub= 2 , lb=1

5. ub=1/100, lb= 200/100−

e 6. ub= 2 , lb=1 7. From the value of integral we get

4=avef . The function f is continuous, therefore f takes on 4 at least once on the given interval.

Evaluate the integrals:

1. dxxx )(

1

0

2+∫ 2. dx

x∫−

−

1

2

2

2 3. ∫ +

π

0

)cos1( dxx

4. dxxx

)11

(4

1

2/1

3−∫ 5. dx

x

x

∫−

4

9

1 6. dxxx 13

3

1

1

2+∫

−

5

7. dttt34

1

0

3 )1( +∫ 8. dttt3/1

7

0

2)1( +∫ 9. dx

x

x

∫− +

0

14

3

9

10. dxx

∫π

03

tan 11. dxx

x

∫−

+

2

1

21

4 12. dx

x

x

∫−

3/

0cos41

sin4π

13. dxe

e

x

x

∫+

4ln

01

2 14. dx

x∫

−

1

0

24

1 15. ∫

+

2

0

228 t

dt

16. dx

xx∫

−+

1

2/12

443

6 17. dx

xx∫

+−

4

2

2106

1 18. xdxe

x

sinh

2ln

0

∫

19. dxx

x

∫4

1

cosh8 20. dx

xx

e

e

∫2

ln

1

Answers:

1. 1 2. 1 3. π 4.-5/6 5. 3 6. 3/22/5 7. 15/16 8. 45/8 9. 2/)103( −

10. 3ln2 11. 2(ln5-ln2) 12. –ln3 13. 2(ln5-ln2) 14. π/6 15. π/16 16. π/2

17. π/2 18. (3-2ln2)/4 19. 16(sinh2-sinh1) 20. )12(2 −

APPLICATIONS of DEFINITE INTEGRALS

Area

1. Find the total area of the region between the curve and the X-axis

a) 23,22

≤≤−−−= xxxy b) 22,43

≤≤−−= xxxy

c) 81,3/1 ≤≤−= xxy

2. Find the area of the region enclosed by the curves

a) 2,22

=−= yxy b) xyxxy =−= ,22

c) xxyxy 4,22+−== d) 2,

2+== yxyx

e) 1,44 42=−=+ yxyx f) 3,2,4

22−=−−=−= xxxyxy

g) 3,2,42

=+−=−= xxyxy h) 2,,/1 === xxyxy

3. Find the area of the region bounded by the curves

a) y = 1/x, y = 5/2 – x b) xxy 22+= , 24 xy −=

c) 2yx = , 4/31 2yx += d) xyxxy 3),1( =−=

e) π/2,sin xyxy == f) 53,)1( 2−=−= xyxy

4. The region bounded below by 2xy = and above by y=4 is to be partitioned into two

subsections of equal area by cutting across it with the horizontal line y=c. Find the value of c.

5. Determine the area of the region enclosed by the Y-axis, the graph of xy = and its tangent

line touching the curve at the point whose abscissa is 4.

6. Find the slope of the line y=mx ( m positive number), if the area of the region enclosed by this

line and the graph of 2xy = is equal to 36.

Answers:

1.a) 28/3 b) 8 c) 51/4 2.a) 32/3 b) 9/2 c) 8/3 d) 9/2 e) 104/15 f) 11/3

g) 11/6 h) 3/2-ln2 3.a) 15/8-2ln2 b) 9 c) 8/3 d) 32/3 e) 1-π/4 f) 1/6

6

4. 23 )4(=c 5. 2/3 6. m=6

Volume

1. Rotate the given curve about the X-axis and determine the volume of the generated solid

a) xy /1= , [ ]3,1∈x b) 2

cos1x

y += , [ ]2/,2/ ππ−∈x c) xxey = , [ ]1,0∈x

2. Rotate the given curve about the Y-axis and determine the volume of the generated solid

a) 2

1

1

xy

+

= , [ ]1,2/1∈y b) xy ln= , [ ]2/3,2/1∈y c) xey = , [ ]2,1∈y

3. Rotate the graph of the function 2/3xy = ( axy ≤≤≥ 0,0 ) about both the Y-axis and the X-

axis. What is the value of a if both solids have the same volume?

Answers:

1. a)3

2π b) )22

4

3(2 +π

π c) )1(4

2−e

π

2.a) )2

12(ln −π b) )(

2

3 ee −π

c) )22ln42ln2( 2+−π 3. a=144/49

Arc length

1. Calculate the arc length of the curve

a) 21,2

1

6

3

≤≤+= xx

xy b) 40,

2/3≤≤= xxy

c) 22,4

1

82

4

≤≤+= yy

yx d) 90,

5

4 4/5≤≤= xxy

2. Find the value of b knowing that the arc length of the curve segment given by the graph of the

function 19

2)(

2/3+= xxf and lying between the points a=0 and b is equal to 42 units.

3. Find the arc length of

a) 2ln2ln,2

≤≤−+

=

−

xee

y

xx

b) dtty

x

∫=0

2cos , 4/0 π≤≤ x

Answers:

1.a) 17/12 b) )110(27

8 2/3− c) 25/16 d) 232/15 2. b=27 3.a) 3/2 b) 1

IMPROPER INTEGRALS

A) Evaluate the integrals:

1. dxx∫∞

1

001.1

1 2. ∫

∞−+

2

24

2

x

dx 3. dx

x∫−

1

1

3/2

1 4. ∫

∞

∞−+

22 )1(

2

x

xdx 5. dx

x

x

∫∞

+0

21

arctan 6. dxxe

x

∫∞−

0

7. ∫−

2

0

24 x

dx 8. ∫

−

2

0 1x

dx 9. dxxe

x

∫∞

∞−

−2

2 10. ∫1

0

ln xdx 11. ∫∞

−2

21

2

x

dx 12. ∫

2/

0

tan

π

xdx

13. ∫∞

++0

2 )1)(1( xx

dx 14. ∫

∞

−++

1

2 65xx

dx 15. dx

x

ex

∫−1

0

7

B) Test the integrals for convergence (integration, comparison test)

1. dxexx/1

2ln

0

2 −−

∫ 2. ∫+

π

0 sin xx

dx 3. ∫

∞

+1

31x

dx 4. ∫

∞

+01 x

e

dx 5. ∫

∞

+06

1x

dx

6. ∫∞

2ln x

dx 7. dx

x

ex

∫∞

1

8. dxx

x

∫∞

+

π

cos2 9. ∫

∞

∞−+14

x

dx

C) Find the values of p for which each integral converges

1. ∫2

1)(ln pxx

dx 2. ∫

∞

2)(ln pxx

dx

Answers:

A) 1. 1000 2. 3π/4 3. 6 4. 0 5. 8/2

π 6. 0 7. π/2 8. 4 9. 0 10. -1

11. ln3 12. +∞ 13. π/4 14. ln2 15. 2(1-1/e)

B) 1. convergent, the value of the integral is 2ln/1−e 2.

xxx

1

sin

10 ≤

+⟨ , dx

x∫π

0

1 is

convergent therefore the original integral is convergent 3. 33

1

1

10

xx

⟨+

⟨ , dxx∫+∞

1

3

1 is

convergent therefore the original integral is convergent 4. x

x

e

e

−

⟨+

⟨1

10 , dxe

x

∫+∞

−

0

is

convergent therefore the original integral is convergent 5. convergent 6. divergent ,

01

ln

1⟩⟩

xx

, dxx∫+∞

2

1 is divergent 7. divergent 8. divergent 9. convergent

C) 1. convergent if p<1 , divergent if p≥1 2. convergent if p˃1 , divergent if p≤1