Thomas Calculus 13th [Solutions]

598 Chapter 8 Techniques of Integration
28.
2
x 6x 6x dx 3
2x dx
dx dx dx dx 3 3 dx
x 3 3 x 3 x 3 x 3 x 3 x 3
2
2 2
2
2
2
2
2
2
2
2
2
2
x
2
1 1 1
tan x 3 3 x 1 tan x
2 2 2
3
3 3 x 3 2
2 3 3 x 2 3
3
C
the first integral we used FORMULA 16 with a 3; for the third integral we used FORMULA 17
with a 3 )
1 1
tan x 3 x
2
2 3 3 x 3
2
2 x 3
C
29.
u x
1 1 1 1 2
sin 1 2 1 1 1 2 1
x dx ; x u 2 u sin u du 2 u 1
1 1 sin u u du u
1 1
sin u u du
2 2
1 u
1 u
dx 2u du
(We used FORMULA 99 with a 1, n 1)
2 1 1 1 1 2 2 1 1 1 2
u sin u sin u u 1 u C u sin u u 1 u C
2 2 2 2
(We used FORMULA 33 with a 1 )
1 1 1 2
x sin x x x C
2 2
30.
u x
1 1
cos x
2 cos 1 1 1 2
dx; x u u 2u du 2 cos u du 2 u cos u 1 u C
x
u
1
dx 2u du
(We used FORMULA 97 with a 1 )
1
2 x cos x 1 x C
31.
u x
2
x 2 1 2 1 2
dx ; x u u 2u du 2 u du 2 1 1
1 2 2
1 1
2 sin u u
2
1 u C sin u u 1 u C
x u u
dx 2u du
(We used FORMULA 33 with a 1 )
1 1 2
sin x x 1 x C sin x x x C
32.
u x
2
2
2 x 2 2 u
2
2
2
2
2
1
dx; x u 2u du 2 2 u du 2 u 2 u sin u C
x
u
2 2 2
dx 2u du
(We used FORMULA 29 with a 2 )
2 1 2 1
u 2 u 2sin u C 2x x 2sin x C
2
2
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