Trigonometric Integrals

Integrals of the form $\int \sin^m x \cos^n x\,dx$ and $\int \tan^m x \sec^n x\,dx$ arise whenever a change of variables (trig substitution, next lesson) converts an algebraic integrand into a trigonometric one. The method is entirely algebraic: split off one factor to serve as the substitute's differential, then use the Pythagorean identities $\sin^2 x + \cos^2 x = 1$ and $1 + \tan^2 x = \sec^2 x$ to rewrite the remaining even powers. The half-angle formulas $\sin^2 x = \frac{1}{2}(1 - \cos 2x)$ and $\cos^2 x = \frac{1}{2}(1 + \cos 2x)$ handle the case when no odd power is available.

• integrate $\int \sin^m x\cos^n x\,dx$ when at least one of $m, n$ is odd;
• integrate $\int \sin^m x\cos^n x\,dx$ when both $m$ and $n$ are even, using half-angle formulas;
• integrate $\int \tan^m x\sec^n x\,dx$ using the identity $1 + \tan^2 x = \sec^2 x$;
• apply reduction formulas for high powers of sine and cosine.

1. Integrals of $\sin^m x \cos^n x$
2. Integrals of $\tan^m x \sec^n x$
3. Standard results worth memorising