Thomas Calculus 13th [Solutions]

CHAPTER 8
TECHNIQUES OF INTEGRATION
8.1 USING BASIC INTEGRATION FORMULAS
1.
2.
0
1
16x
dx
2
8x
2
2
u 8x 2 du 16x dx
u 2 when x 0, u 10 when x 1
1 10
16x
1
10
dx du ln u
2 2
0 8x
2 2 u
x
x
2
2
ln10 ln 2 ln5
dx
1
Use long division to write the integrand as 1
x
x
2
1
dx 1 dx dx
1 x 1
2 2
1
x tan x C
x
1
2
.
1
3.
2
sec x tan x dx
Expand the integrand:
2 2 2
sec x tan x sec x 2sec x tan x tan x
2 2
sec x 2sec x tan x (sec x 1)
2
2sec x 2sec x tan x 1
2 2
sec x tan x dx 2 sec x dx 2 sec x tan x dx 1 dx
2 tan x 2sec x x C
We have used Formulas 8 and 10 from Table 8.1.
4.
/3
1
2
/4 cos x tan x
dx
2 1
u tan x du sec x dx dx
2
cos x
u 1 when x 4, u 3 when x / 3
/3 3
1 1
3
dx du ln u
2 1
/4 cos x tan x 1 u
1
ln 3 ln1 ln3
2
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