Page 224 - Calculus Demystified

CHAPTER 7
Methods of Integration
EXAMPLE 7.11 211
Calculate the integral

2
cos xdx.
SOLUTION
Of course we will use formula II. We write

1 + cos 2x
2
cos xdx = dx
2

1 1
= dx + cos 2xdx
2 2
x 1
= + sin 2x + C.
2 4
EXAMPLE 7.12
Calculate the integral

3 2
sin x cos xdx.
SOLUTION
When sines and cosines occur together, we always focus on the odd power
(when one occurs). We write
3 2 2 2 2 2
sin x cos x = sin x sin x cos x = sin x 1 − cos x cos x

2 4
= cos x − cos x sin x.
Then
3 2
2 4
sin x cos dx = cos x − cos x sin xdx.
A u-substitution is suggested: We let u = cos x, du =− sin xdx. Then the
integral becomes
3 5

2 4 u u
− u − u du =− + + C.
3 5
Resubstituting for the u variable, we obtain the final solution of
5
3
cos x cos x
3 2
sin x cos dx =− + + C.
3 5
You Try It: Calculate the integral

2
5
sin 3x cos 3xdx.