Solucionario larson 8 edicion

108

Chapter 7

83. For n

Integration Techniques, L’Hôpital’s Rule, and Improper Integrals

1,
x

I1

x2

0

dx

4

1

b

1
2

lim

b→

x2

1

4

2x dx

1
1
6 x2 1

lim

b→

0

b
3

1
.
6

0

For n > 1,
x2n

In

2

x

0

x 2n

u

2,

1

1

n

du

0

1

4

dx

1

5

dx

1

x2

6

x
0

x20

2

x

85. False. f x

1
4

b

1

x
x2

1

2

x

0

5

1

x
1

x2

1

x3

2
5

dv

6 x2

0

4

3

0

0

3

11
46

dx

dx

n

n
n

2

n

dx, v

1
2

2n

x2n
0

2

x

3
n

1

1
2 x2

1

n

2

dx

n
n

0

1
I
2n

2

1
24

21
5 24

1
60

, lim 1 x

b

1

0, but

1

x→0

x

dx

1

lim

b→

ln x

1

87. True

Review Exercises for Chapter 7
x x2

1
2

1 dx

x2

1

1 x2 1
2
32
12
x
3

5.

ln 2x
dx
x

ln 2x
2

1

12

3.

2x dx

13
9

e2x sin 3x dx
e2x sin 3x dx

(1) dv
u

1
2x
dx
2 x2 1

dx

1
ln x2
2

C
32

C

7.

1 2x
e cos 3x
3

2
3

2 1 2x
e sin 3x
33

1 2x
e cos 3x3

16
dx
16 x2

16 arcsin

2 2x
e sin 3x
9

e2x
2 sin 3x
13
v

⇒ du

1

C

C

2

sin 3x dx ⇒
e2x

1

32

1 2x
e cos 3x
3

e2x sin 3x dx

9.

x
x2

e2x cos 3x dx

3 cos 3x
1
cos 3x
3

2e2x dx

2
3

e2x sin 3x dx

C
(2) dv
u

cos 3x dx ⇒
e2x

v

⇒ du

1
sin 3x
3
2e2x dx

.
0

Diverges

1.

1

1
6

1 iscontinuous on 0,

1x

b

x2n 2
2 x2 1

2n

3 dx,

lim

x5

(c)

b→

b→

3

(b)

lim

2 x 2n

2n

x

(a)

dx

3

x
4

C

Review Exercises for Chapter 7

11. u

x, du

dx, dv

x

5 dx

2
xx
3

5

2
x
3

2
x
3

5

4
x
15

5

32

2
xx
3

xx

v

1 2 dx,

5

5

32

x

5

32

2
x
3

4
x
15

x

532

6
x
15

4
3

5

32

3x

1
8
11
82

1
8x2
16

u

⇒

x dx

1

4x2

2x 1

4x2

2x

arcsin 2x

2
4x2

dx

sin2

x

1 cos

sin

x

1

13
sin
3

x

1

C

x

13

sin2

x

1

C

x

13

1

cos 2

1
sin
3

x

12

cos 2

x

1 dx

1
1

19.

sec4

x
dx
2

tan2
tan2

x
2

21.

1

1
sind

1

1 sec2

x
x
sec2
dx
2
2

2 3x
tan
3
2
sin
cos2

2 tan

d

1
sin 2x
2
dx

C (by Formula 43 of Integration Tables)

C

1
sin
3

x

2x dx

dx

1
sin
3

17.

1
cos 2x
2

2 2x 2
dx
1
2x 2

1 arcsin 2x

1

v

⇒ du

x

x2
2

v

arcsin 2x ⇒ du

cos3

4x2

1

x2
arcsin 2x
2

dv

cos 2x dx ⇒

C

x2

x2arcsin 2x
2

v

⇒ du

x2

u

C

x
sin 2x
2

sin 2x dx ⇒

(1) dv

u

x2
arcsin 2x
2

x arcsin 2x dx

C

1
x sin 2x
2

12
x cos 2x
2

C

5

x cos 2x dx

12
x cos 2x
2

3 2 dx

52

10

12
x cos 2x
2

x2 sin 2x dx

13.

(2) dv

2
x
15

15.

32

5

x
2

x

1 dx

x
1

1

C

C

x
dx
2
sec2
C

sec2

x
dx
2x
2
tan3
3
2

sec

tan

3 tan

x
2

C

d

tan

sec

C

109

1
2

sin 2x dx

1
cos 2x
4

C

110

Chapter 7

23.

Integration Techniques, L’ H ô pital ’s Rule, and Improper Integrals

12
dx
x2 4 x2

24 cos d
4 sin2 2 cos

2
x

csc2

3
3 cot

2 sin , dx

25.

x

4

2 sec2

x2

4 − x2

x2

2 cos

C

d,

x2

4

2cos

2 tan

dx

θ

C

34
x
x

d

4 sec2

x2 + 4

d

x

θ
2

x3
dx
4 x2

8 tan3
2 sec

2 sec

8 tan3

sec d

8
8
8

2

sec2

1 tan sec d

sec3
3
x2

sec

12 2
xx
3

4

x2 dx

x2
2

12
x
3

4

12
x
3

C

32

4
24

x2

27.

d

4
8
3

4
4

12

2 cos

4

x2

4

x2

C

4

8

2 cos

C

C...