Integrals of Trig Functions

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Integrals of Trig Functions

Antiderivatives of Basic Trigonometric Functions
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Product of Sines and Cosines (only even powers)
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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.