Integrals From R to Z

8/10/2019 Integrals From R to Z

1/20

2

2

1

0

3

3

3

4

0

1

10

32

6

21

ln

2

2

0

2

3

1

1

1

1

4

1

ln

sin cos

ln

log

sin

ln 1

.

.

.

.

.

.

.

.

.

.

.1

n

x

ne

x

e

x

x

x xI dx

x

x

I dxx

xI dx

x

I dxx

I x dx

I x dx

I dxx x

I x x dx

xI dx

e

I x dx n

I x a dx

I x dx

I x dx

I dx dxx

I x e dx

xeI d

x

x

2

2

2

0

2

1 32

1

3 2

2 2

4

43/4

2

1

3

1

ln

1

2 7

1

1

1

sin cos

2

2 2

1

1

ln

.

.

.

.

3

.

.

.

.

.

3

1

1 1

1 .

.

.

2

ln

ln

3

x

x

x x

n

e

xI dx

x

xI dx

x x

eI dx

e

xI dx

x

I x x x dx

I e dx

x xI dx

x

I dxx x

xI x dx

I x x dx

xI dx

x x

xI dxx

ax

I e dx

I x dx

xI x e

0

dx

Easy as

2

2

2

cos

0

2

2sin

22

0

1

1

0

1

0

1

0

1

1

1sin

!

1

4 8

sin

sin 1

sin 2

ln

1

1

si

1

.

1

n .

x

x

x

x

a

a

x

x

ne

x

I dxx

ex

I dxx

eI dx

x e

xI dx

x x

I x e dx

I x dx

I x e dx

I e dy

xx eI e dx

x

x

zI dz n

z

I dx

I dxx x

I e x dx

I dxx

I

1

0

1lnx dx

x


2/20

2

2

1

0

1

3

0

2 2

2

2

2

2

ln 1

1

2 3

1

1

2

2 2

1

tan

1

ln

2 1

1

1

1

2

2 8

.

ln

1

ln,

x

x

x

n

x x

x x

x x

x x

e e

x

x

e

e x

n

I dx

I x e dx

x

I dx

I nx dx

I dxx a

I x dx

e eI dx

e e

e eI dx

e e

x xI dx

x x

I dxe

I x x e dx

I dx

x x

I dxx x x

I e dx

xI d

x

x

x

x

n

2

2

12

2

2

1/ 2 2sin cos

0

ln 1

2 3

1

/3

0

ln

1

2

2

3 2

2 4

0

1

2

3

ln

1ln

2

1

2

16

1

cos1

2 sin 1

ln 2

2 3

5 6

2

ln

1 1sin

ex x

x

ex

x x

x

x

ex

I dx

e

I e e dx

eI dx

x

x xI dx

x

xI dx

x

I dx

x xI dx

x x

I e dx

I x x dx

eI dx

x

I dx

I x dx

Ixx

5

4 22

2/2

4 2

1

sin 0.5

dx

I dxx x

I dxx

1

2

2

/2

/6

3

1

0

2

1

3/ 2 3

4 41/2

1 2

3 20

23

6

sin 2

2sin

ln

1 2

cos sin cos

1

3

1

tan

1

1

ln

1

1

5 5

8 8

2 5

6 11 6

1 cos 2

4

e

x

xI dx

x

xI dx

x

I x x dx

I x x dx

xI dx

x x

I dxx

I dxx x

I dxx x x

xI dx

x

I x e dx

xI dx

x x

x xI dx

x x x

xI

1

6

4 3

1

1

1

cos sin 2

.

n

dx

I dx nx x

xI dx

x x x

I x x dx


3/20

1: Show that the area under the function f x a x is equal to58

15a when a is a

positive integer.

2: Solve the following integral by substitution 2

3 ln 2 ln4

x x dxx

3: Find the area bound by 2f x x , g x x n and the x-axis in the domain 0, where n is a positive integer.

Bonus question: find the area bound by f x , g x and the y-axis in the same domain.

4: Find the area under the function 24f x x by graphing. What can you say about the

area of 2 2g x r x ? What volume do we get if we rotate 360g x around the x-axis?

5: Let g x x and 2f x x . We start measuring the area under g x and f x from 0to a . Assume that the area under g(x) equals the area under f(x). How large is this area?

6. Given that 5 ,. . 5B A

C B

f x dx f x dx and . 0A

C

f x dx

What can you say about f x ?Can you think of any possible functions having these properties?

7. Find a such that equals zero.

8. Prove or disprove the following inequality 14 14 14b a cI I I

where

1 1

0 0

. .1 1

,11

A B

xI dx I dx

xx

and1

01

.Cx

I dxx

.

9. Show that 5

3 1x x dx equals 61

3 11 121

x x C by substitution.

10. Use two different methods to integrate x xh x a e with respect to x . Where a is an

integer and xe denotes the exponential function.

11. Prove the following statements: (These will become very useful for the harder integrals)

0

.

a a

b

f x dx f a b x dx

0 0 0

. .a b a a a a

b a a

f x dx f x dx , f x dx f a x dx , f x dx f x f x dx

0

ln

a

x dx


4/20

12. Solve the following integral 23 23 3 6 1x x x x dx by substitution.

13. Find the integral of224A xe xe dx dx

where A is defined as 22

1

xdx

x .

14. Evaluate

4

0

2 2 1.I x dx by drawing

15. Find the area between nA x x and nB x x when n is a positive integer.


5/20

14. Is 13 larger, equal, or less than

4

0

.x x dx ?

15. The expression

0

3

3

1 1!2!3!4!

2 1 2

dx

x

can be simplified to ab

a b. Evaluate

16. Find A and B such that the integral 2 6.

B

A

x x dx achieves its maximum value.

17. Find the area between sinf x x and sing x x from 0 to n where n is a positive integer. Now rotate the functions 180 degrees around the x-axis, what is the

volume from 0 to n ?

18. Evaluate1/P

e where P is defined as

1 1

0

1 2

2 ln 22

x

x dx

19. (Hard) Find the area of the moustache (red) where sinf x x and cosg x x Also find the area of the grey Area.

20. Find Ge where G is given by1

20

1

p xdx

p x

and p is an positive integer.

21. Find the area bounded by 1

lnf xx

, 0x and y n where n is a positive integer

22. A function F x is defined as1 3

x

x. Show that

23 2 1 3

27F x dx x x C

Evaluate 1/3

0

F x dx . Hint this is a improper integral.

23. Integrate3

21

x

x using two different methods. 24. Evaluate

2 ln n

dx

x x

where n is a

positive integer greater than one.

25. Find the area restricted by the x-axis and where n is a odd integer and

2 2a b

1nn xf x x e

1n


6/20

25. (Easy one) Find the area restricted by the functions: sin , sin , 2y x x y y x (The hat shape )

26. Find the green area expressed by the chord A.

27. Consider the function 2f x x from 0 to a.

How much bigger is the area of 2F than 1F ? (Find the ratio between 2F and 1F). Consider

the function ng x x . What is the ratio between 2F and 1Fnow ?


7/20

Medium rare

1

4 4

3

2

2

3

2

3 21

7

2

cos sin

sin cos

sin ln

1 1

1 1

sin cos

cos sin

2sin

cos

2 1

1

1 1

2ln 2

ln 1

1

1

1 2

1

.

.

.

.

.

.. .

.

. .

.

e

x

x

x xI dx

x x

I x dx

x xI dx

x x

x xI dx

x x

x

I dxx

x eI dx

x

I dx

x x

xI dxx

xI dx

x

I dxx x

xI dxx

eI

e

ln 2

0 1

..

.

xdx

x kI dx

x

2

0

2 2

4

22cos 1

0

2

9

4

5

7

1

0

2

2

. . . ... .

.

. .

.. .

.

.

. .

ln

1 1

1

sin 4cos

1

1

1 tan

1

3 2

7

2

ln 1 1

1 cos

2

sin 2

.

.2 sin

1 cos

ee

x

x

xI dx

x

x xI dxx

I x x e dx

I dxx x

I dxx

xI dx

x x

xI dx

x

I x x dx

I x dx

I dxx

xI e dx

x

I

22

1 3

30

1 2

1

1

ln

t

.

a .n

x

x xe

x e dx

xI dx

x

eI x dx

x

I x dx

2

2

2

20

99

2

2

4

0

1

0

2

.

.

.

.

1

, 0

1

1

2 3

9 4

sin 101 sin

1

2 5

1

2 4

4sin 3cos

1

ln ln ln

cos

cos sin2 2

p

x x

x x

x

dtI

x t x t

I d

x pI dx p

x p

xI dx

x

I dx

I x x dx

I x dx

dxI

x x

I dx

dxI

x x

xI dx

x

I x x dx

xI

x x

3

2

35 2

1

2

0

1 2

1

arcsin

1

c.

o

.

s

dx

xI dx

x x

I x dx

I dxx


8/20

/2

0

2 2

2

0

22

20

1 1ln ln

2 2

0

11

0

20

1/4

0

2

4 32

2

1 sin

1

2 5

2

sin

1 sin

ln , 0

, 2ln

4

1 1

ln l

x

x x

n

p

dxI

x

dxI

x x

dx

I x x

I dx

xI dx

x

dxI

e e

dxI

x x n

I x x dx n

dxI

x x

I x xdx

dxI px x

I x dx

Ix

2

22

n

1

dxx

dxI

x

0

2

0

2

2

2 2

1

2

2

0 4 3

22

3

2 2

2

4

4

2

2 1

sin cos

5

6 13

1 ln ln 1 1

1,

2

5 3 5 2

1

1

5 4

2 3

2

4

1

1

1

sin 2

x x

x

x

a

x x

dxI

e e

I e dx

I x e x dx

xI dx

x x

dxI

x x x

dxI a

x a

dxI

x x

x xI dx

x

x x x

I dxx x

e x eI dx

x

xI dx

x x

dxI

x x

I

3 5

22 11

2 2 10

ln tan

cos sin

2 1 cot

1 1 1 cot

x x dx

dxI

x x

x x xI dx

x x x

2 2

2

2

2

2010

4 1/7

1/7 1/73

3

/2

1

2

4 3 2

ln

1

1

log

2

cos sin

1

7

csc sin

1ln ln

ln

cossin ln

, 21

1 cot

1 cot

1

2 3 2 1

cos sin

x

xa

e x

n

xdx

x

dx

x

x e x dx

xdx

x x x

dx

x x

xdx

x x

xe dx

x x dx

dx

x x

x dx

x

xx x dx

x

dxn

x x

x

dxx

xdx

x x x x

x x

2

cos

arctan

arctan

xdx

x x x

x

dxx x


9/20

2

0

1/3

1

1

1 2

cot tan

'''

sec tan

sin cos

sin sin 2

sinh cosh

2 9 9

3

sin cos

1 sin 2

cosh sinh

cosh sinh

sin arccos

1 ln

csc sin

sec tan sin

x

x x dx

f x f x dx

dx

x x

x x dx

x x dx

x x dx

x xdx

x x

x xdx

x

x xdx

x x

x dx

x x dx

x x dx

dx

x x x

/4

0

2 1 1

2 2

1

2

23

21

tan cos 2 tan

1 2

ln 1

sec tan 2

1 ! 2 1 !

2 2 1 ! !

xx x

dx

x x x

x x dx

e x x dx

x x x x dx

p pdp

p p

2

220

2 2

21/ 2

40

3 2 /23

2/32

0

1

0

21

2

22

2

22

0

2

3/22

1

3

ln

arcsin

1

4 9

1 cos 2

ln 1

1

1

1

1

arctan

1

2 4

3

xe e x

n

k

t t

x

xe

xdx

x x

x xdx

x a

x x dxx

xdx

x

x dx

e dx

dn

xdx

x x

xdx

x

tdt

e t e

x xdx

x

n

k

dx

x x

3

,32

dx

n ndn n

12

5

2

0

/2

0

ln 4

2

ln 4/3

/2

1 arctan

21

/2

/6

1/2 2

20

ln 10 ln 2

0

2 20

4

cos sin

sin 2

2 2 cos

16

1 sin cos

1

tan sin

8

1 2

1

3 24

, 0

arctan

x x

x

x

x x

x

a

xI dx

x

I x x dx

xI dx

x

I e e dx

dxI

x x

eI dy

y

xI dx

x x

xeI dx

x

e eI dx

e

dxI a

x a

I

3

1

1

0

20

sin 3 sin

cos cos 3

cos 3 cos

1 ln

ln

x

e

xdx

x

x xI dx

x x

dxI dx

x x

I x e xdx

xI dx

x e


10/20

1: Show that the indefinite integral arccos arcsin .x x dx equals 2x C

.

Find the area under the function arccos arcsinf x x x

2:Show that the area under the function ln k xf xk x

where f is bounded by its

asymptotes, is given by 4 l n 2k , where k is an integer.

3: Calculate the area between 1

nf x x and g x x . What happens as limn

?

4: Use the following integral

41 4

20

1

1

x xdx

x

to prove that

22

7

.

5: Calculate the integral

1

sin ln

n

ex

dxx

where n is a positive integer.

11/2. ComputeA

e where A is defined as

4/3 2

3 23/4

2 1

1

x xdx

x x x

6: Given the integral

2

22 2

xdx

x a

show that it is equal

2a

when a is not equal to zero.

7: Solve the following integral . . . ... . . . .2 2 4 2 2 4 ..I x x x x dx by clever

factorization. Hint: First show that 2

2 2 4 2 2x x x

8: The integral 2

3 2

6

sin 2 cos 3 .x x dx

can be written asb

a

b

. Find 1b aa b .

9 a) Find the area bounded by nxf x e , n

g xxe

and the x-axis.

Where n is a positive integer. What is the area when 2n ?

9 b) Bonus question: Find the area bounded by f x , g x and the y-axis. When n>1. What happens to the area as n approaches infinity?

10: Find the n`th integral of the following functions.

1 1

, , , ,x

x

a x

er x e p x f x x g x h x

xx a


11/20

11. Let a be a positive real number. Find the value of a, such that the definite integral2a

a

dx

x x Achieves its smallest possible value

12. Evaluate ln B where B is defined as

1

3 611 1

dx

x x .

13 EvaluateT

e where T is given byln 2 3 2

3 20

2 1

1

x x

x x x

e edx

e e e

14 Let 2 31 ...2 4 8x x xf x for 1 1x . Find We where

1

0

W f x dx

13. The integral 332 998 1664 6911

6661

2 4 sin

1

u u u udu

u

can be written as

1

2 2

aa

b a

find

15. Show that the integral

23

2 2

1

2 2 11 ln x xxx x dx

evaluates to 13 .

16. Find the area enclosed by ln af x x , 0x and the tangent that goes through thepoint ,a f a where a is greater than zero.

17. Find the area between sin sin 2 sin 3T x x x x and the x-axis from 0 to .

18. Find the area of the ellipse:

a b


12/20

19. Solve the following integral if possible22

4

2 1 4

xdx

xxe xe dx dx

20: Consider the regionRunder the restricted by the functions:

; 0 ; 0 , 0 , 0x x x xf x e x , g x e x , h x e x , k x e x

Find the volume of the solid obtained by revolving this region 180 degrees around they-axis.

21: Find the volume of a regular cone whose height ishand the radius of its circular base is r

by integration.

22: Evaluate

2

2 2

6sin cos sin 2 23sin

cos 1 5 sin

x x x xdx

x x

23: Consider the finite three-sided regionGbounded by the graphs of

Find the area of G. Find the volume of G by revolving this region about the y-axis.

24: Show that the integral 92 3 2z z dz can be written as

25: Find the surface of a torus (a "tire") with radius of a cross-section equal torand the

radius of rotation of the centre of cross-section equal toR.

26: Find the area restricted by b blog , log and 0y x y x x where b is the positive

base. For example 10log ln , log 100 2e x x .

27: Evaluate the definite integral

3 2/2

2/4

cos cot csc

sin tan sec

x x xdx

x x x

28. Consider the triangle formed by the x- and y-axes and the line tangent to

2

11f x

x

at the point ,a f a , where 0a . For what value of a is the area of the triangle smallest?

39. Find the value of 0

sinrxe rx dx

when r is a positive integer.

30. Find the integral of the following functions:

1

2 1 2 11 sin sin

dxI

x x x x

22 , 2 , 6 3y x y x y x

1021 220 60 9 2 35280

z z z C

2 3cos sin sin cos sin cos cosh ln cosh tanhx

I x x x dx I x x x x dx


13/20

31. Let A equal 2 2

cos 2 sin 2 cos sina x a x a x a x . Evaluate 2

1

0

a

A x dx

32. (Hard) Ignore the constant for a minute, now

Let x

nxF n dx

e Calculate

3 1

2

21

F dxF

e dx

and

3 1

1

12

F dxF

e dx

33. Find the integral of sin ln cos lnI x x dx using two different methods.

34. Show that

29

172

0 5 16

xdx

x

equals 2 15

14!

2 49 5 16!


14/20

, , !Fun fun fun

Which integral can i take?

50

0

1

21

0

1

20

0

21 1

0

1

0

2

0

20

1 2

20

ln 1

1

1

sin

sin

1 2 cos

1

ln

ln 1

1

tan cot

ln 1

ln sin

sin

1 s

.

.

.

.in

ln

1.

.

.1

x

x

I x x dx

xI dx

x

xI dx

x

I dx

x x

xI dx

x

xI dx

x

I x e dx

I x x dx

xI dx

x

I x dx

x xI dx

x

x xI dx

x

xI

1 1

2 20 0

2

..

.

cos

ln

1

1

dxx

x xyI dy

x yxd

2

0

2011

1

1

0

2

0

20

0

4

2

1

0

2

0

2

0

0

3

ln

ln ln 1

1

1 tan

1 cos

sin cos

sin

ln 9

ln 3 ln 9

1

1ln

si

. ..

.

. .

n

1 sin

1

.

.

.

x

x

x

x

I e dx

xI dx

x

I x x dx

I dxx

xI dx

x

x xI dx

e x

xI dxx x

I dx

x

I x dx

xI dx

x

xI dx

e

eI

4

0

23

0

1

.

. t ..an

xe

dx

x

I x dx

22

2 20

/2

0

1

0

2cos

0

0

21

20

3/433 4

0

22

0

2

2

1

1 1

sin

sin cos

sin ln

ln

cos sin

ln 1

1

17 403

5 2 2

ln

1

sin

1

c

. ... .

os

4

x

xx

dx dxI and Ix x x x

xI

x x

xI dx

x

I e x dx

I e dx

xI dx

x

x xI dx

x

xI dx

x

x xI dx

ex

xI d

x

2

2

32

2

0

5 50

/2

0

2

20

2

2

cos 32

1

sin

cot

ln 1

1

sin

x

ex

x xI dx e

x

e xI dx

x

dxI

x x

I x x dx

xI dx

x

xI dx


15/20

1

0

3

20

22

20

21

2 20

24

0

1

0

1

20

/4

/4

0

1

0

1

0

ln 1 ln 1

sin

1

cos cos 3

arctan 2

1 2

3

ln ,

ln

4

1 cot

sin tan 11

2

11 ln 1

1 sin ln

ln

x

a

x

x

x x dx

xdx

x

dx

x e

x xdx

x

xdx

x x

dx

x a

xdx

x x

xdx

x

xdx

x e

x e xdxe

x x xdx

x

/3

0

cos0

12

0

4

sinlim

12sin cos

1 sin

sin ln

n

n nn

x

xdx

x x

dx

x

x dx

0

1

20

1

0

0

2 2cos

1

ln

1

ln

1

2 2cos

x

x

xI dx

xe

x xI dx

x x

xI dx

x

xI dx

xe


16/20

Solve the following integrals where every letter except x, denotes a positive integer.

1

10

2 20

0

220

2 20

1

0

2

10

0

! 1ln

1

ln 1ln

2

1 cos

1 cos

1

11

1

4

, , 1ln

cos cos2

.

1 1

.

.

.

.

nnm

n

n

a

a b

n

n

nax x

nI x x dx

m

xI dx n

nx n

nxI dx n

x

nI= dx

nx x

I dx

x a x

x xI dx a b

x

I x nx dx ,n,m N

I e e dx

2

0

0

10

/2 2008

2008 2008/2

322 2

0

! 1 !

!

2 31 8

41

ln 1

1 sin

!

sin

41 2007 sin cos

1ln sin ln 2

24 2

.

.

1

.

n

k k

ax

p

a bx

a

x

n a

a n

kI dx C

x

I e dx

xI dx C

x p

aI x e dx where n,m N

b

xI dx

x x

I

k

x dx

2

2 20

222

0

2

0

1

2 20

6

2 4

2

1 1, 0

4cos sin

ln 1 cos cos 12 0

cos 8

1 1ln 1 cos 2 ln

2

sin ln cos ln 1 2arctan

ln 2 1

1

1

x x

x x

dxI a b

a baba x b x

xI dx

x

aI a x dx

p x q x pI euler!

x p p

e eI

x e e e

6 80

1 12

0

cos

0

2 12 1 2 2

3 5ln

2

ln0 1 cot csc

1

cos sin 0

2, , 0

2

1 1sin arctan 1

1ln

1

x x

a

a x

nna

x

x

e

x xI dx where a a a

x

I a x e dx , a a

ndxI n N a

n ax x a

I x dxx e

eI

e

0dx


17/20

0: Remember the formulas you showed in the last easy problems. Namely 10.

1/2: Show that the integral

0

1

1.

nI dx

x

equals to cscn n

when n is a positive integer.

(1/2)! Show that

1

0

1! ln

x

x dtt

use this to prove that

1 1!

2 2

1: Prove that if f is continuous on ,a b and that f x f a b x is constant for all

then the integral b

a

f x dx equals (Useful trick)

: Ifh is a bounded nonnegative function, that exists prove that

2: For 0a , prove that

2

2

4cos

a

xe ax dx e

3: Prove that the integral

20sin 1

sin 1

.ax

dx

bx x

is equal to

sinh

2 sinh

a

b

if a b .

4: Solve

23 3 3 3

0

. . ... . .2 2 1 10 6 .1 ..I x x x x dx

5: Show that

1

2 2

sin cossec sin 2 1

sin cos sin cos. . .

. ..sin cos

x xdx x C

x x x x x x

6: Compute the integral

4

6 1.1xI dxx

. Give the answer in the form arctan Q x CP x

where ,and P x Q x x

7: Evaluate the following limit 2

0

lim 1 sinn

nn dxx

,x a b

0

1 ln0

xh x dx

x x

12 2

a bb a f a f b b a f


18/20

8: Show that 20

ln 1 2 cos 0 1,1a x a dx if a

else 2 ln a

9: For a positive integer n, compute the integral21 ...

!

.

2!

n

n

xdx

x xxn

10 a) : The function :f is given by

20

13 .

sin

2

xt

f x dtt

Find without, solving the integral a polynomial 2p x ax bx c such that

10 b) P x is a third degree polynomial with two distinct roots. The integral between the

roots are equal to 2 7 / 4 . Find P x .

11: Give an example of a function : 2, 0,f with the property that

2

pf x

is finite if and only if 2,p

12: Find f x when arctan

2

0

ln 1

x

f t dt x

13: Determine all continuous functions : 0,1f that satisfy 1

0

1

12f x x f x dx

14: Find the integral of 2 2

2 cos 2 2 sin 2x ye x xy y xy dy

14: Find all continuous functions : 0,1f satisfying 1 1

2 2

0 0

1

3f x dx f x dx

15: Let n be an odd integer greater than 1. Determine all continuous functions : 0,1f

such that 1

1/

0

. . . . . . .1 , 2 , .... , 1 .. .n x

k kf x dx , k n

n

16: Find the maximal value of the ratio

33 3

3

0 0

/f x dx f x dx

as f ranges over all positive continuous functions on 0,1 .

0 0 0 ' 0 0 '' 0p f , p' f and p'' f


19/20

17: Findn such that 2

10

cos 0n

k

kx dx

18: Compute 1

1

arccos nx dx

when n is a odd integer.

19: A square hole of side length 2b is cut symmetrically through a sphere of radius a. Where

2a b . Find the volume removed.

20: Find the derivative of

cos2

sin

cosh

x

x

t dt

21: Find the area restricted by the x-axis and the function cos lnf x x for 0, 2x

22: Putnam(A1 93)

The horizontal liney =c intersects the curve 32 3y x x in the first quadrant as in the

figure.Findc so that the areas of the two shaded regions are equal.

23: (Putnam 85 5B) Evaluate 1 120

1985t tt e dt

you may assume that2x

e dx

.

24: (Putnam 91 5B) Find the maximum value of 2

4 2

0

y

x y y dx for 0 1y

25: Putnam(A5 93) Show that111

2 2 210 2 2 21011

3 3 31 101100

101 1003 1 3 1 3 1

x x x x x xdx dx dx

x x x x x x


20/20

is a rational number.

26: Putnam(A3 97) Evaluate3 5 7 2 4

2 2 20

... 1 ...2 2 4 2 4 6 2 2 4

x x x x xx dx

27: For what pairs ,a b of positive real numbers does the improper integral

0

x a x x x b dx

converge?

28: Let n and 1

2

n nt

where t Show that the integral

1

0

sin sin 2 sin sinn n

x x nx txI dx

x

equals

!

2

n

29: If f is a continuous function on 0,1 such that 1

12

f x f x

evaluate 1

0

f x dx

30: Evaluate the integral

1

0

1 1 1x x x dx

31; Evaluate ln R wherek x

eR dx

k x

, k

Integrals From R to Z

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