Page 226 - Calculus Demystified

Methods of Integration
CHAPTER 7
Integrals involving the other trigonometric functions can also be handled with 213
suitable trigonometric identities. We illustrate the idea with some examples that are
handled with the identity
2
2
2
sin x sin x + cos x 1
2 2
tan x + 1 = + 1 = = = sec x.
2
2
2
cos x cos x cos x
EXAMPLE 7.14
Calculate

3
3
tan x sec xdx.
SOLUTION
Using the same philosophy about odd exponents as we did with sines and
2
2
cosines, we substitute sec x − 1 for tan x. The result is

2 3
tan x sec x − 1 sec xdx.
We may regroup the terms in the integrand to obtain

4 2
sec x − sec x sec x tan xdx.

A u-substitution suggests itself: We let u = sec x and therefore du =
sec x tan xdx. Thus our integral becomes
5 3
4 2 u u
u − u du = − + C.
5 3

Resubstituting the value of u gives
5 3
3 3 sec x sec x
tan x sec xdx = − + C.
5 3

EXAMPLE 7.15
Calculate
π/4

4
sec xdx.
0