Thomas Calculus 13th [Solutions]

596 Chapter 8 Techniques of Integration
9.
2 3
2 2 ( x 2)(2x 3 2) 2 2 x x 2 1
x 4x x dx x 2 2x x dx sin x 2 C
6 2 2
2 2
( x 2)(2x 6) 4 x x 1 2 ( 2)( 3) 4
1
4sin x
x x x x
C
4sin x 2
6 2 3 2
(We used FORMULA 51 with a 2 )
C
10.
x x
x
2
1 2
x x
1
2
x
2
x
1
2
2
1 2 1 1 2 1 1
dx dx 2 x x sin C x x sin (2x 1) C
2 2 2
(We used FORMULA 52 with a 1 )
2
11.
dx
dx
2 2
x 7 x x 7 x
2
2
7 7 x
1 ln
C 1 ln
7 x
7
2
7 7
x
x
2
C
(We used FORMULA 26 with a 7 )
12.
dx
dx
2 2
x 7 x x 7 x
2
2
7 7 x
1 ln
C 1 ln
7 x
7
2
7 7
x
x
2
C
(We used FORMULA 34 with a 7 )
13.
2 2 2 2 2 2
4 x 2 x 2 2 2 2 x 2 2 4 x
dx dx 2 x 2ln C 4 x 2ln C
x x x x
(We used FORMULA 31 with a 2 )
14.
15.
2 2 2
x 4 x 2 2 2 1 2 1
dx dx x 2 2sec x C x 4 2sec x C
x
x
2 2
(We used FORMULA 42 with a 2 )
2t
2t
2t e cos3 t dt e (2cos3 t 3sin 3 t ) C e
2 2
2 3
13
(2cos3 t 3sin 3 t ) C
(We used FORMULA 108 with a 2, b 3 )
16.
3t
3t
2 2
3t e sin 4 t dt e ( 3sin 4 t 4cos 4 t ) C e
( 3) 4
25
( 3sin 4 t 4cos 4 t ) C
(We used FORMULA 107 with a 3, b 4 )
17.
18.
1 1 1 1 2 2
1 1 1 1 1
x cos x dx x cos x dx x 1 1
1 1 cos x x dx x
1 1 2 2 cos x x dx
2 2
1 x
1 x
(We used FORMULA 100 with a 1, n 1)
2 2
x 1 1 1 1 1 1 2 1 1 1 1 2
cos x sin x x 1 x C x cos x sin x x 1 x C
2 2 2 2 2 2 4 4
(We used FORMULA 33 with a 1 )
1 1 1 1 2 2
1 1
2 2 2
1 1 1 1 1
x tan x dx x tan (1 x ) dx x
1 1 tan (1 x ) x dx x
1 1 1 (1) 2 tan x x dx
x
2 1 x
(We used FORMULA 101 with a 1, n 1)
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