7.4. Trigonometric Integrals http://www.ck12.org
7.4 Trigonometric Integrals
Learning Objectives
A student will be able to:
- Compute by hand the integrals of a wide variety of functions by using the Trigonometric Integrals.
- Combine this technique withu−substitution.
Integrating Powers of Sines and Cosines
In this section we will study methods of integrating functions of the form
∫
sinmxcosnxdx,wheremandnare nonnegative integers. The method that we will describe uses the famous trigonometric identities
sin^2 x=^12 ( 1 −cos 2x)and
cos^2 x=^12 ( 1 +cos 2x).
Example 1:
Evaluate∫sin^2 xdxand∫cos^2 xdx.
Solution:
Using the identities above, the first integral can be written as
∫
sin^2 xdx=∫ 1
2 (^1 −cos 2x)dx
=^12∫
( 1 −cos 2x)dx
=^12 (x−^12 sin 2x)+C
= 2 x−^14 sin 2x+C.Similarly, the second integral can be written as