Thomas Calculus 13th [Solutions]

Section 8.2 Integration by Parts 563
49.
1
u sec t , du dt ; dv t dt , v t
2
1
2
;
t t
2
2 2 2
1 1 2 2
2
2 t dt
t sec t dt t sec t
t dt 2
2 3 2 2 3 2 2 3 3 6 2 3 2
2 3
t t 1 2 t 1
2
5 1 2 5 1 4 5 1 3 5 3 5 3 3
t 1 3 1 3
9 2 9 2 3 9 2 3 9 3 9
2 3
50.
1 2 2x dx
2
u sin x , du ; dv 2 x dx, v x ;
4
1 x
1 2 1 2
1 2 2 1 2 1 2 2 2x dx
1 2 4
1/2
1 1
3
2x sin x dx x sin x x 1 x 4x dx
0 1 x
2 6 2
0 0 4
0
1 2
4 3 6 3 12
1 x
1
12 12 4 12
0
51.
x tan
1 1 1
x dx Let u tan x, du dx, dv
2
1 x
x dx,
v
x tan
1
x dx
2
1 2 1 1 x
x tan x dx
2 2 2
1 x
1 2 1 1 1
x tan x 1
2 2 2
1 x
dx
1 2 1 1 1 1
x tan x x tan x
2 2 2
C
1 2 1 x
x 1 tan x
2 2
C
2
x
2
52.
3
2 1 x 1 x 1/ 2
2 x
x tan dx Let u tan , du dx, dv x dx,
v
2 2 2
1 ( x / 2)
3
1 3
3
x
2 1 x x 1 x 1
x tan dx tan 2 dx
2 3 2 3 2
x 1
4
3
x 1 x 1 2x
tan 2x
3 3 3
2
x
1
4
dx
3
x 1 x 1 2 1 2x
tan x
3 2 3 3 2
x
1
4
2
x x
In the remaining integral, let w 1 , dw .
4 2
dx
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