Thomas Calculus 13th [Solutions]

Section 8.2 Integration by Parts 555
49.
7 4
x x 1 dx
4 3 7 u 1
u x 1, du 4 x dx;
x dx du
4
7 4 1
x x 1 dx ( u 1) u du
4
1 3/2 1 1/2
u du u du
4 4
1 5/2 1 3/2
u u C
10 6
1 3/2 1 4
3/2
4
u 3u 5 C x 1 3x 2 C
30 30
50.
2
2/3
( x 1)( x 1) dx
x 1 2
The easiest substitution to use is probably u , du dx.
x 1 2
(1 x)
The integral can be written as
1 1 2/3
dx u du
2/3
x 1
2
2
( x 1)
x 1
1/3
3 1/3 3 x 1
u C C
2 2 x 1
8.2 INTEGRATION BY PARTS
1.
u x, du dx; dv sin x dx, v 2cos x ;
2 2
xsin x dx 2x cos x 2cos x dx 2x cos x 4sin x C
2 2 2 2 2
2. u , du d ; dv cos d , v 1 sin ;
cos d sin 1 sin d sin 1 cos C
2
3. cost
2
t
( )
sin t
2t
( )
cost
2
( )
sin t
0
2 2
t cos t dt t sin t 2t cost 2sin t C
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