What if the integral involves secant and tangent?

Use the identity $ an^2(x) + 1 = \sec^2(x)$. If the power of secant is even, save $\sec^2(x)$ and let $u = an(x)$. If the power of tangent is odd, save $\sec(x) an(x)$ and let $u = \sec(x)$.

How do I handle definite integrals of trigonometric products?

Change the limits of integration when performing the $u$-substitution to avoid converting back to $x$, or compute the indefinite integral first and evaluate it at the original bounds.

What if both powers are odd?

You can choose either substitution. Usually, factoring out the term with the smaller odd power requires less algebraic expansion.

How to integrate a trigonometric product

Integrating trigonometric products of the form $\int \sin^m(x) \cos^n(x) \, dx$ requires trigonometric identities and $u$ -substitution. It applies when at least one power is a positive integer, or when both are even non-negative integers. If one power is odd, save one factor and convert the remaining even powers using the Pythagorean identity. If both are even, use half-angle identities to reduce the degree of the integrand.

The setup

Identify the exponents $m$ and $n$ in the integral $\int \sin^m(x) \cos^n(x) \, dx$ . Determine if at least one is odd, or if both are even. Choose the appropriate trigonometric identities: use the Pythagorean identity ( $\sin^2(x) + \cos^2(x) = 1$ ) to convert even powers when an odd power is present. Use the half-angle identities ( $\sin^2(x) = \frac{1 - \cos(2x)}{2}$ and \cos^2(x) = \frac{1 + \cos(2x)}{2}) when both powers are even to step down the exponents.

The steps

If $m$ is odd, save one $\sin(x)$ factor for $du$ and use $\sin^2(x) = 1 - \cos^2(x)$ to express the remaining sine factors in terms of cosine. Let $u = \cos(x)$ . 2. If $n$ is odd, save one $\cos(x)$ factor for $du$ and use $\cos^2(x) = 1 - \sin^2(x)$ to express the remaining cosine factors in terms of sine. Let $u = \sin(x)$ . 3. If both $m$ and $n$ are even, apply half-angle identities to reduce the powers until all terms can be integrated directly. 4. Integrate the resulting polynomial in $u$ or the reduced trigonometric terms. 5. Substitute the original variable back and add the constant of integration $C$ .

Checking the result

Differentiate the computed antiderivative using the chain rule and the product rule. Simplify the resulting expression using algebraic expansion and the Pythagorean identity. The fully simplified derivative must exactly match the original integrand.

Common errors

A frequent error is forgetting the negative sign when substituting $u = \cos(x)$ since $du = -\sin(x) \, dx$ . Students also often expand binomials incorrectly when converting higher powers, such as $(1 - u^2)^2$ . When both powers are even, failing to properly apply the half-angle identity, particularly mixing up the internal signs for sine and cosine, will ruin the integration.

Worked example

Evaluate $\int \sin^3(x) \cos^2(x) \, dx$ .

The power of sine is odd ( $m=3$ ). Save one $\sin(x)$ factor: $\int \sin^2(x) \cos^2(x) \sin(x) \, dx$ . Convert $\sin^2(x)$ to $1 - \cos^2(x)$ : $\int (1 - \cos^2(x)) \cos^2(x) \sin(x) \, dx$ . Let $u = \cos(x)$ , so $du = -\sin(x) \, dx$ . Substitute $u$ : $\int (1 - u^2) u^2 (-du)$ . Distribute the negative sign and $u^2$ : $\int (u^4 - u^2) \, du$ . Integrate with respect to $u$ : $\frac{u^5}{5} - \frac{u^3}{3} + C$ . Substitute back $u = \cos(x)$ : $\frac{\cos^5(x)}{5} - \frac{\cos^3(x)}{3} + C$ .

FAQ

Run your own problem

Answer form

References: Stewart Calculus, 8th Edition · OpenStax Calculus Volume 2, Chapter 3 · Khan Academy: Trigonometric integrals