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

Graphical or geometric meaning

On the unit circle, sine and cosine are linked by $\sin^2+\cos^2=1$. Saving one derivative factor lets the remaining even power be projected entirely onto the other coordinate.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate using product and chain rules.
• Use identities to compare apparently different antiderivatives.
• Check parity strategy before expanding powers.

Parity determines the first identity

For powers of sine and cosine, save one sine factor when its power is odd and convert the rest using $1-\cos^2x$; do the symmetric operation for an odd cosine power. When both powers are even, use half-angle identities. For tangent and secant, an even secant power often reserves $\sec^2x$, whereas an odd tangent power may pair with $\sec x\tan x$ after conversion. The saved factor must create a derivative pair, not merely follow a mnemonic. Keep definite bounds consistent and watch the minus sign in a cosine substitution. Differentiate the antiderivative, using identities to recover the original integrand, before trusting a complicated simplification.

Mixed products with different angles may instead call for product-to-sum identities. First ask whether all factors share one angle and whether a single substitution will remove the remaining trigonometric powers; parity rules are less useful when those structural conditions fail.

Practice

1. Evaluate $\int\sin x\cos^2x dx$.
2. Rewrite $\sin^2x$ for an even-power integral.
3. Evaluate $\int\sec^2x dx$.

Connections and next steps

• Substitution Rule
• Trigonometric Substitution
• Integration by Parts