
## Section1.8Trigonometric Integrals

Integrals of polynomials of the trigonometric functions $\sin x\text{,}$ $\cos x\text{,}$ $\tan x$ and so on, are generally evaluated by using a combination of simple substitutions and trigonometric identities. There are of course a very large number  1 The more pedantic reader could construct an infinite list of them. of trigonometric identities, but usually we use only a handful of them. The most important three are:

Notice that the last two lines of Equation 1.8.3 follow from the first line by replacing either $\sin^2x$ or $\cos^2x$ using Equation 1.8.1. It is also useful to rewrite these last two lines:

These last two are particularly useful since they allow us to rewrite higher powers of sine and cosine in terms of lower powers. For example:

\begin{align*} \sin^4(x) &= \left[ \frac{1-\cos(2x)}{2} \right]^2 &\text{by Equation }\knowl{./knowl/eq_TRGINTtrigidentityF.html}{\text{1.8.4}}\\ &= \frac{1}{4} - \frac{1}{2} \cos(2x) + \frac{1}{4}\underbrace{\cos^2(2x)}_{\text{do it again}} & \text{use Equation }\knowl{./knowl/eq_TRGINTtrigidentityG.html}{\text{1.8.5}}\\ &= \frac{1}{4} - \frac{1}{2} \cos(2x) + \frac{1}{8}\left(1 + \cos(4x) \right)\\ &= \frac{3}{8} - \frac{1}{2} \cos(2x) + \frac{1}{8}\cos(4x) \end{align*}

So while it was hard to integrate $\sin^4(x)$ directly, the final expression is quite straightforward (with a little substitution rule).

There are many such tricks for integrating powers of trigonometric functions. Here we concentrate on two families

\begin{align*} \int \sin^mx \cos^nx \dee{x} &&\text{and}&& \int \tan^mx \sec^nx \dee{x} \end{align*}

for integer $n,m\text{.}$ The details of the technique depend on the parity of $n$ and $m$ — that is, whether $n$ and $m$ are even or odd numbers.