Thomas Calculus 13th [Solutions]

Chapter 8 Additional and Advanced Exercises 657
34. 3x
e ( ) sin 4x
3
3 x
e ( )
1
cos 4x
3
9e x ( )
1
sin 4x
16
4
3 x 3 x 3 x 3
e e e e
x
3 9 25
4 16 16 16 16 25
I cos 4x sin 4x I I 3sin 4x 4cos 4x I 3sin 4x 4cos 4x C
35. sin 3x ( ) sin x
3cos3x ( ) cos x
9sin 3x ( )
sin x
sin 3 cos
3cos3 sin
8
I sin 3x cos x 3cos3xsin x 9I 8I sin 3xcos x 3cos3xsin x I
x x x x
C
36. cos5x ( ) sin 4x
5sin 5x ( )
1
cos 4x
4
25cos5x ( )
1
sin 4x
16
1 5 25 9 1
5
4 16 16 16 4 16
1
4cos5 cos 4
9
5sin 5 sin 4
I cos5x cos 4x sin 5xsin 4x I I cos5x cos 4x sin 5xsin 4x
I x x x x C
37.
ax
e ( ) sin bx
ax
ae ( ) 1 cos bx
2 ax
a e ( ) 1 sin bx
2
b
b
ax ax 2 2 2 ax ax
e ae a a b e e
b 2 2 2 2 2 2
b b b b a b
I cosbx sin bx I I a sin bx bcos bx I asin bx b cosbx C
38. ax
e ( ) cosbx
ax
ae ( ) 1 sin bx
2 ax
a e ( ) 1 cosbx
2
b
b
ax ax 2 2 2 ax ax
e ae a a b e e
b 2 2 2 2 2 2
b b b b a b
I sin bx cosbx I I a cosbx bsin bx I a cosbx bsin
bx C
39. ln ax ( ) 1
1
x
( ) x
1
x
I x ln ax x dx x ln ax x C
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