Answers to Selected Exercises in Chapter 6

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Section 6.2

Section 6.3

Section 6.4

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

-1/2

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

19. ò₀^p/3( sin( 2x) -sin( x)) dx+ò_p_/3^p( sin( x) -sin(2x) ) dx = 5/2

21. ò₀^{ln( 4)}( 5-e^x+4e^-x) dx = 10ln( 2)

23. ò₀^p/4( secx-tanx) dx = ln( Ö2+1) -ln( 2) /2

25. omit

27. ò₀^p/6( sec²( x) -2sec( x)tan( x) ) dx+ò_p_/6^p/4( 2sec(x) tan( x) -sec²( x) ) dx = 1-2[Ö3]+2Ö2

29. ò₁^¥( x^-2-x^-3) dx = 1/2

Section 6.2

5. V = pò₀¹(3x+2)²dx = 13p

7. V = pò₀³( x+1) dx = 15p/2

9. V = pò₀^psin( x) dx = 2p

11. intersect at x = 0 and x = 1: V = pò₀¹( ( [Ö(x)]) ²-( x²) ²) dx = 3p/10

13. omit: typographical error

15. V = pò₀^p/4( 2cos²( x) -1)dx = pò₀^p/4cos( 2x) dx = p/2

17. V = 2pò₀¹xÖ{1-x²}dx = 2p/3

19. revolve f( x) = Ö{R²-x²} over [ 0,R] about the y-axis;

V = 2p

ó
õ

æ
Ö

R²-x²

dx =

2pR³3

21. Revolve f( x) = bÖ{( 1-x²/a²) } over [ -a,a] about x-axis:

V = pb²

ó
õ

-a

æ
ç
è

x²a²

ö
÷
ø

dx =

b²a

Section 6.3

5. L = ò₀¹Ö{1+( 3²) }dx = [Ö10]

7. L = ò₀¹Ö{1+( ³/₂x^1/2) ²}dx = ( 13[Ö13]-8) /27

9. change interval to [ p/6,p/2] :

L = ò_p_/6^p/2Ö{1+( cotx) ²}dx = ln(4)

11. interval is [ 0,1] : will use identities (6.15) and (6.16)

L =

ó
õ

æ
Ö

æ
ç
è

x^1/2-

x^-1/2

ö
÷
ø

dx =

13. will use identities (6.15) and (6.16):

L =

ó
õ

æ
Ö

æ
ç
è

e^4x-

e^-4x

ö
÷
ø

dx =

e⁴-

e^-4-

15. will use identities (6.15) and (6.16):

L =

ó
õ

ln( 2)

æ
Ö

æ
ç
è

e^2x-e^-2x2

ö
÷
ø

dx = -

e²+

e^-2+

17. change interval to [ 1,2] :

L =

ó
õ

æ
Ö

1+( x³-1)

dx =

Ö2-

19. change interval to [ 0,p/3] :

L =

ó
õ

p/3

æ
Ö

1+( sec⁴( x) -1)

dx = Ö3

21. the arclength is

L =

ó
õ

p/3

p/6

æ
Ö

1+( cot( 2x) ) ²

dx = ln( 4)

21. since f'( x) = x², we have

L =

ó
õ

æ
Ö

1+x⁴

dx = 1.0894294132

23. since f'( x) = ( x+1) ^-1/2/2, we have

L =

ó
õ

æ
Ö

4( x+1)

dx = 1.0830642795

25. since f'( x) = e^x, we have

L =

ó
õ

æ
Ö

1+e^2x

dx = 2.0034971116

27. Since F'( x) = e^{-x^2/2}, we have

L =

ó
õ

æ
Ö

1+e^{-x^2}

dx = 1.3194285585

Section 6.4

_
x

ó
õ

x[ f( x) -g( x)] dx,

_
y

ó
õ

( f(x) ²-g( x) ²) dx

5. Area is A = ò_-1³( 2x+3-x²) dx = 32/3:

_
x

ó
õ

-1

x( 2x+3-x²) dx = 1,

_
y

ó
õ

-1

( ( 2x+3) ²-x⁴)dx =

7. Area is A = ò₀¹( x-( 2x²-x³) )dx = 1/12 :

_
x

= 12

ó
õ

x( x-( 2x²-x³) ) dx =

_
y

= 6

ó
õ

( x²-(2x²-x³) ²) dx =

9. Area is A = ò_-1¹( x²+1-2x²) dx = 4/3:

_
x

ó
õ

-1

x( x²+1-2x²) dx = 0,

_
y

ó
õ

-1

( ( x²+1)²-( 2x²) ²) dx =

11. Area is A = ò_-1¹( x⁴+1-2x²) dx = 16/15:

_
x

ó
õ

-1

x( x⁴+1-2x²) dx = 0,

_
y

ó
õ

-1

( ( x⁴+1)²-( 2x²) ²) dx =

13. Area is A = 2ò₀¹( x³-x⁵) dx = 1/6:

_
x

ó
õ

-1

x( x⁵-x³) dx+6

ó
õ

x(x³-x⁵) dx = 0

_
y

ó
õ

-1

( x¹⁰-x⁶) dx+3

ó
õ

(x⁶-x¹⁰) dx = 0

15. Area is A = 2ò₀¹( 2-x²-x) dx = 7/3:

_
x

ó
õ

-1

x( 2-x²-| x| )dx,

_
y

ó
õ

-1

( ( 2-x²)²-x²) dx =

17. Area is A = 2ò₀^p/4( sec²( x) -tan²( x) ) dx = p/2:

_
x

ó
õ

p/4

-p/4

x( sec²(x) -tan²( x) ) dx = 0,

_
y

ó
õ

p/4

-p/4

( sec⁴( x) -tan⁴(x) ) dx =

19. omit because of cosh(x):

21. Area is A = ò₀^Ö{p}sin( x²) dx » 0.8948

_
x

0.8948

ó
õ

Ö{p}

xsin(x²) dx » 1.1176,

_
y

2(0.8948)

ó
õ

Ö{p}

sin²(x²) dx » 0.3743

23. Area is upper half os unit circle of radius 1: A = p/2

_
x

ó
õ

-1

æ
Ö

1-x²

dx = 0,

_
y

ó
õ

-1

( 1-x²) dx =

25. Area is A = ò₁⁴( e^5x-4-e^{x^}²) dx = 627,822.3885

_
x

627,822.3885

ó
õ

x(e^5x-4-e^{x^}²) dx = 3.68,

_
y

2( 627,822.3885)

ó
õ

(e^10x-8-e^{2x^2}) dx = 2,293,694.5107

Section 6.5

d
dx

tan^-1( 3x) =

1+9x²

d
dx

tan^-1( e^x) =

e^x1+e^2x

d
dx

tan^-1( Öx) =

2Öx( x+1)

11.

d
dx

Öxtan^-1( Öx) =

2Öx

tan^-1( Öx) +

2( x+1)

13.

d
dx

sin( 2tan^-1( x) ) = 2

cos( 2tan^-1( x) )

1+x²

= 2

x²-1

( 1+x²) ²

15.

ó
õ

dx
x²+16

tan^-1

æ
ç
è

x
4

ö
÷
ø

17.

ó
õ

e^xdx
e^2x+1

= tan^-1( e^x) +C

19.

ó
õ

xdx
x⁴+3

pÖ3

21.

ó
õ

xtan^-1( x) dx =

x²tan^-1( x)-

tan^-1( x) +C

23.

cos( tan^-1( x) ) =

æ
Ö

1+x²

25.

cos( 2sin^-1( x) ) = 1-2x²

27.

sin( tan^-1( x²) ) =

x²

æ
Ö

1+x⁴

Section 6.6

d
dx

sin^-1( 3x) =

æ
Ö

1-9x²

d
dx

sin^-1( e^x) =

e^x

æ
Ö

1-e^2x

d
dx

sin^-1( Öx) =

2Öx

___
Ö1-x

11.

d
dx

( cos^-1( x) +sin^-1( x)) = 0

13.

d
dx

sin( 2sin^-1( x) ) = 2

cos( 2sin^-1x)

æ
Ö

1-x²

= 2

1-2x²

æ
Ö

1-x²

15.

ó
õ

æ
Ö

5-x²

= sin^-1

æ
ç
è

xÖ5

ö
÷
ø

17. omit: have not covered these functions yet

19.

ó
õ

( 2x+1) dx

æ
Ö

1-( x²+x) ²

d
dx

sin^-1( x²+x) +C

21. multiply top and bottom by e^x:

ó
õ

æ
Ö

e^-2x-1

ó
õ

e^x dx

æ
Ö

1-e^2x

= sin^-1( e^x) +C

23. consider that x = ( Öx) ²if x > 0. Thus,

ó
õ

Öx

æ
Ö

1-( Öx) ²

= 2sin^-1( Öx) +C

25. (this one is not easy!)

ó
õ

xsin^-1( x) dx =

x²sin^-1( x) -

sin^-1( x) +

æ
Ö

1-x²

27.

sin( cos^-1( 2x) ) =

æ
Ö

1-4x²

29.

tan( sin^-1( x) ) =

æ
Ö

1-x²

31.

cos( 2sin^-1( x) ) = 1-2x²

Section 6.7

ó
õ

1.5

x
9-x²

dx = -

ln3+ln2

ó
õ

2Ö3

x²

æ
Ö

16-x²

dx = -2Ö3+

ó
õ

Ö3

æ
Ö

x²+3

= ¥

11.

ó
õ

-1

æ
Ö

1-x²

dx =

13.

ó
õ

[Ö15]

dx
( x²+5) ^3/2

Ö3

15.

ó
õ

x²

æ
Ö

4+x²

= -

æ
Ö

4+x²

17.

ó
õ

dx
( 25-x²) ^3/2

æ
Ö

25-x²

19.

ó
õ

Öx

æ
Ö

1-x²

dx =

ó
õ

p/2

æ
Ö

sin( q)

dq = 1.19814

21.

ó
õ

æ
Ö

16-x²9-x²

dx =

ó
õ

p/2

æ
Ö

16-9sin²( q)

dq = 5. 27388 8432

23.

ó
õ

æ
Ö

x( 9-x²)

ó
õ

p/2

æ
Ö

3sin( q)

= 1. 51384 56348

Section 6.8

5. 2xsinh( x²)

7. 3x²cosh( x³)

9. tanh( x)

11. e^xsech( x) -e^xtanh( x) sech²( x)

13. 2cosh( x) sinh( x)

15. 6x²sinh( x³) cosh( x³)

17. cosh( px) /p+C

19. òx²cosh( x) dx = x²sinhx-2xcoshx+2sinhx+C

21.

ó
õ

sinh²( x)

cosh³( x)

dx = -

sinhx
2cosh²x

+tan^-1( e^x) +C

23.

ó
õ

dx
tanh( x) +sech( x)

ó
õ

cosh( x) dx
sinh( x) +1

= ln( sinhx+1) +C

25.

ó
õ

cosh( x)

dx = 2tan^-1( e^x) +C

27. 5.1983485413

29.

ó
õ

sin

æ
ç
ç
ç
ç
è

æ
Ö

x²-4

ö
÷
÷
÷
÷
ø

dx =0.62724 57965 1