Integrals of Powers of Sine and Cosine

Use substitution when at least one exponent is odd. Factor out one term from the odd-powered function, rewrite the remaining even-powered term using the Pythagorean identity in terms of the other function, then choose the appropriate variable for substitution. This transforms the integral into a polynomial that can be integrated directly.

First, factor out one term from the odd-powered function. Second, rewrite the remaining even-powered term using a trigonometric identity in terms of the other function. Third, choose the appropriate function as the substitution variable, find its derivative, and integrate the resulting polynomial expression before converting back to trigonometric terms.

When both exponents are even, direct substitution is ineffective. Half-angle identities replace sine squared and cosine squared with equivalent expressions, reducing the powers and simplifying the integrand. This transformation makes the integral easier to solve step by step into a simpler form.

The Pythagorean identity converts even powers of one trigonometric function into expressions involving the other function. This allows substitution to work effectively by transforming the integral into a polynomial. For example, sin²(x) = 1 - cos²(x) enables cosine substitution when sine has an odd power.

In a purely resistive AC circuit, average power consumed by a resistor is found by integrating voltage and current. The integral involves even powers of sine and cosine, so half-angle identities are applied to solve it. This demonstrates how trigonometric integrals simplify real-world engineering calculations.

The decision depends on whether the exponents are odd or even. If one power is odd, use substitution with the Pythagorean identity. If both powers are even, apply half-angle identities as reduction formulas to simplify the integrand before integration. These techniques are foundational to integrals of powers of secant and tangent.

Selecting the appropriate substitution variable—typically the function whose derivative appears in the integrand—transforms the trigonometric integral into a polynomial form. The separated factor from the odd-powered function combines with dx to form the differential, enabling direct polynomial integration and simplifying the overall solution.