Answers to Selected Exercises in Chapter 5

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Section 5.3 Section 5.4 Section 5.5 Section 5.6 Section 5.7 Section 5.8 Section 6.1

Section 5.3

Comment: There is a typographical error at the beginning of this exercise set.

ò13xdx = 4 should be ò13f(x)dx = 4. Also, ò23xdx = 2.5 should be ò23f(x)dx = 2.5.

5. 0

7. 4+7 = 11

9. -ò24g(x)dx = -7

11. ò12g(x)dx = ò14g(x)dx+ò42g(x)dx = ò14g(x)dx-ò24g(x)dx = 5-7 = -2

13.
 
ò02f(x)dx = ò01f(x)dx+ò12f(x)dx
2 = ò01f(x)dx+1.5 

So ò01f(x)dx = 0.5

15.12

17. 2·2-3·4 = -8

19. 6

25. ò13-5xdx = -ò135xdx = -20 (Area of trapezoid with base lengths 5 and 15 and height 2)

29. =5ò0Ö3Ö{3-x2}dx =5·3/4p = [15/4]p  (five times the area of a quarter-circle of radius Ö3.)

31.=ò1Ö3xdx+ò0Ö3Ö{3-x2}dx = 3/2+3/4p


Section 5.4

5. 3x

7. x2

9. [(d)/(dx)]òx-11/tdt = [(d)/(dx)]-ò-1x1/tdt (from property 5.19 on page 330.)

= -1/x

11. sin(z2)

13. [1/(Ö{cosh2x-1})][(d)/(dx)]cosh(x) = [1/(Ö{cosh2x-1})]sinh(x)

15. eln(x)[(d)/(dx)]ln(x) = x·1/x = 1

17. [1/(Ö{1-sin2(x)})][(d)/(dx)]sin(x) = [1/(Ö{1-sin2(x)})]cos(x)


Section 5.5

5. ò024x3dx = 16

7. ò-125x5dx = 105/2

9. ò-ppcos( 3x) dx = 0

11. ò15x-1dx = ln5

13. ò19x-1/2dx = 4

15. ò0.51.5e7xdx = 83.1981746

17. ò-ppsin( x3) dx = 0

19. ò-22( 4x2-cos( px) ) dx = 64/3

21. ò01( ex+e-x) 2dx = e2/2-e-2/2+2

23. ò0p/4cos( 2x) dx = 1/2

25. ò14[(Öx+x3/2)/( Öx)]dx = 21/2

27. 0

29. ò02x2dx = 8/3

31. ò01e3xdx = (e3-1)/3

33. 1

35. ò0p/4sec( x) dx = ln( Ö2+1)

37. ó
õ		 1

0

 xdx
1+x
= 
ó
õ		 1

0

 1+x-1
1+x		 dx
= 
ó
õ		 1

0

 1+x
1+x		 dx- ó
õ		 1

0

 dx
1+x
= 
1-ln2

39. ¥

41. ò02psin( x/2) dx = 4


Section 5.6

5. ò01xe-xdx = -2e-1+1

7. ò0pxsin( 2x) dx = -p/2

9. omit

11. ò1exln( x) dx = ( e2+1) /4

13. omit

15. ò02x2e-2xdx = ( 1-13e-4) /4

17. ò01xln( x+1) dx = 1/4

19. ó
õ		 5

0

   ____
Ö3x+1		 dx = 14
21. ó
õ		 3Ö{p}

0

 x2sin( x3) dx =  2
3
23. ó
õ		 1

0

 xdx
  æ
Ö
4-x2
= 2-Ö3

25. omit

27. ò-ppesin( x) cos( x) dx = 0

29. omit

31. ó
õ		 1

0

 zdz
  æ
Ö
2-z2
= Ö2-1

33. ò01[(ex-1)/( ex-x)]dx = ln( e-1)

35. ó
õ		 2

1

 ln( Öx) dx
x		 =  1
4		 ln2( 2) 


Section 5.7


5. does not exist

7. ò-11x-2/5dx = 10/3

9. does not exist

11. does not exist

13. does not exist

15. omit

17. ò0¥xe-xdx = 1

19. ó
õ		 ¥

1

 1
x2		 dx = 1
21. ó
õ		 ¥

0

 e-x
e-x+1		 dx = ln2

23. diverges

25. omit
 


Section 5.8


5. ò06x/6dx = 3

7. 0

9. ò0¥pxe-pxdx = [1/( p)]

11. ó
õ		 ¥

0

 1
2		 x3e-x = 3

13. omit

15. 0

17. ó
õ		 6

0

 ( x-3) 2
6		 dx = 3,    s = Ö3
19. ó
õ		 ¥

0

 p æ
ç
è		 x- 1
p		 ö
÷
ø		 2
 		 e-pxdx =  1
p2		 ,    s = 1/p


Section 6.1


5. ò-13( 2x+3-x2) dx = 32/3

7. ò01( x-( 2x2-x3) ) dx = 1/12

9. ò-11( x2+1-2x2) dx = 4/3

11. ò-11( x4+1-2x2) dx = 16/15

13. 2ò01( x3-x5) dx = 1/6

15. 2ò01( 2-x2-x) dx = 7/3

17. ó
õ		 -1/2

-2

 ( | x| -( x+1) )dx+ ó
õ		 1

-1/2

 ( ( x+1) -| x| ) dx = 7/2

19.

ò0p/3( sin( 2x) -sin( x)) dx+òp/3p( sin( x) -sin(2x) ) dx = 5/2

21. ò0ln( 4) ( 5-ex+4e-x) dx = 10ln( 2)

23. ò0p/4( secx-tanx) dx = ln( Ö2+1) -ln( 2) /2

25. omit

27.

ò0p/6( sec2( x) -2sec( x)tan( x) ) dx+òp/6p/4( 2sec(x) tan( x) -sec2( x) ) dx = 1-2[Ö3]+2Ö2

29. ò1¥( x-2-x-3) dx = 1/2