Trigonometric Integrals

The central idea

Integrals of powers of sine and cosine use parity. For $\int\sin^m x\cos^n xdx$, save one sine factor when $m$ is odd and convert the rest with $\sin^2x=1-\cos^2x$; save one cosine when $n$ is odd. If both are even, use half-angle identities. Tangent–secant powers have analogous derivative pairings.

Definitions, hypotheses, and notation

For $\int\tan^m x\sec^n xdx$, an even positive secant power suggests saving $\sec^2x$ and substituting $u=\tan x$; an odd tangent power suggests saving $\sec x\tan x$ and using $u=\sec x$. Identities $\sec^2=1+\tan^2$ then convert remaining factors.

Some integrals allow several correct routes and yield antiderivatives that look different but differ only by a constant. Differentiation and trigonometric identities, not visual similarity, establish equivalence on a connected domain interval.

Conceptual meaning

The strategy manufactures an inner function together with its derivative. Identities convert leftover even powers into the chosen substitution variable. Parity predicts which derivative factor can be reserved.

A dependable method and decision rule

1. Classify the trig functions and exponent parity.
2. Reserve one factor matching the derivative of the intended substitution.
3. Rewrite remaining squared factors with Pythagorean identities.
4. Substitute and integrate the resulting polynomial.
5. Return to the original variable and differentiate to check.

Fully worked example

Common mistakes and why they fail