Page 223 - Calculus Demystified

Methods of Integration
CHAPTER 7
210
Now we resubstitute the x-expressions to obtain

cos x
dx = log | sin x|+ C.
sin x
Finally we can evaluate the original definite integral:
π/2
π/2
cos x
dx = log | sin x|
π/3 sin x π/3
√
3
= log | sin π/2|− log | sin π/3|= log 1 − log
2
1
=− log 3 + log 2.
2
You Try It: Calculate the integral
3

tdt
.
2
2
−2 (t + 1) log(t + 1)
7.4 Integrals of Trigonometric Expressions

Trigonometric expressions arise frequently in our work, especially as a result of
substitutions. In this section we develop a few examples of trigonometric integrals.
The following trigonometric identities will be particularly useful for us.

I We have
1 − cos 2x
2
sin x = .
2
The reason is that

2
2
2
2
2
cos 2x = cos x − sin x = 1 − sin x − sin x = 1 − 2 sin x.
II We have
1 + cos 2x
2
cos x = .
2
The reason is that
2
2
2
2

2

cos 2x = cos x − sin x = cos x − 1 − cos x = 2 cos x − 1.
Now we can turn to some examples.