Integrals of Trig Functions

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Integrals of Trig Functions

Antiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (only even powers)
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Order of Integration

Product of Sines and Cosines (only even powers)

If $n$ and $m$ are both even, then the tricks of the previous slide do not help. Instead, we can either integrate by parts (using the "go in a circle" trick in the previous module) or use double-angle formulas. Most people find the double-angle formulas to be easier, and that's what this slide's video is about.

Remember the trig identities we said would be helpful:

$\sin^2(x) = \frac{1-\cos(2x)}{2}$

$\cos^2(x) = \frac{1+\cos(2x)}{2}.$ This converts

$\sin^n(x)\cos^m(x)$ , with

$n$ and

$m$ even, into terms of

$\cos(2x)$ . The odd powers of

$\cos(2x)$ can be handled as in the previous slide. To integrate the even powers of

$\cos(2x)$ , apply the double-angle trick again, getting a polynomial in

$\cos(4x)$ . Repeat as many times as necessary. While this can be tedious, it is not hard.