Integrating Powers of Sine & Cosine

Integrals involving powers of trigonometric functions cannot be computed using anti-derivative formulas or the Substitution Rule.

What follows are strategies and specific techniques for integrating these types of functions.

Most of the techniques involve using trigonometric identities to algebraically transform the integrand into something more easily integrated. Therefore, the following identities will be useful:

Pythagorean Identity

$\cos^2(x) + \sin^2(x) = 1 \\$
$\sin^2(x) = 1 - \cos^2(x) \\$
$\cos^2(x) = 1 - \sin^2(x) \\$
$\sec^2(x) = 1 + \tan^2(x) \\$
$\csc^2(x) = 1 + \cot^2(x)$

Fundamental & Reciprocal Identities

$\csc(x) = \frac{1}{\sin(x)} \\$
$\sec(x) = \frac{1}{\cos(x)} \\$
$\tan(x) = \frac{\sin(x)}{\cos(x)} = \frac{1}{\cot(x)} \\$
$\cot(x) = \frac{\cos(x)}{\sin(x)} = \frac{1}{\tan(x)}$

Half-Angle Identities

$\cos^2(x) = \frac{1}{2}\left(1+\cos(2x)\right)$

$\sin^2(x) = \frac{1}{2}\left(1-\cos(2x)\right)$

Double-Angle Identities

$\sin(2x) = 2\sin(x)\cos(x) \\$
$\cos(2x) = \cos^2(x) - \sin^2(x) \quad = 2\cos^2(x) - 1 \quad = 1 - 2\sin^2(x)$

Negative-Angle Identities

$\sin(-x) = -\sin(x)$

$\cos(-x) = \cos(x)$

Integration Techniques

Odd Powers of Sine & Cosine

With odd powers of $\sin$ and $\cos$, you split off a single power of the function, rewriting, for instance, $\cos^m(x)$ as $\cos^{m-1}(x) \cdot \cos(x)$:

$$ \require{color} \int \cos^5(x) dx \quad = \quad \int \cos^4(x) \cdot \cos(x) dx \quad \textcolor{gray} { Split \space off \space one \space power} \\ = \int \left(1 - \sin^2(x)\right)^2\cos(x) dx \quad \textcolor{gray} {Pythagorean \space Identity} \\ = \int (1 - u^2)^2 du \quad \textcolor{gray} {Let \space u = \sin(x). \space du = \cos(x)dx} \\ = \int (1 - 2u^2 + u^4) du \quad \textcolor{gray} {Expand} \\ = u - \frac{2}{3}u^3 + \frac{1}{5}u^5 + C \quad \textcolor{gray} {Integrate} \\ = \sin(x) - \frac{2}{3}\sin^3(x) + \frac{1}{5}\sin^5(x) + C \quad \textcolor{gray} {Replace \space u \space with \space \sin(x)} $$

Even Powers of Sine & Cosine

With even powers of $\sin$ and $\cos$, use the Half-Angle Identities to reduce the powers in the integrand:

$$ \int \sin^4(x) dx = \int \left( \sin^2(x) \right)^2 dx \quad \textcolor{gray} {Rewrite \space to \space make \space next \space step \space more \space obvious} \\ = \int \left(\frac{1}{2}(1 - \cos(2x)) \right)^2 dx \quad \textcolor{gray} {Half-Angle \space Identity} \\ = \int \frac{1}{4}\left(1 - 2\cos(2x) + \cos^2(2x)\right) dx \quad \textcolor{gray} {Expand} \\ = \frac{1}{4}\int \left(1 - 2\cos(2x) + \cos^2(2x)\right) dx \quad \textcolor{gray} {Move \space constant \space out \space of \space integrand} \\ = \frac{1}{4}\int \left(1 - 2\cos(2x) + \frac{1}{2}\left(1+\cos(2\cdot2x\right)\right) dx \quad \textcolor{gray} {Use \space \textbf{Half Angle Formula} \space again} \\ = \frac{1}{4} \int \left(\frac{3}{2} - 2\cos(2x) + \frac{1}{2}\cos(4x)\right) dx \quad \textcolor{gray} {Simplify} \\ = \frac{3x}{8} - \frac{1}{4}\sin(2x) + \frac{1}{32}\sin(4x) + C \quad \textcolor{gray} {Evaluate \space the \space integrals} $$

Example 1

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

$$ \int \sin^3(x) dx = \int \sin^2(x) \cdot \sin(x) dx \quad \textcolor{gray} {Split \space off \space one \space power} \\ = \int (1 - \cos^2(x))\sin(x)dx \quad \textcolor{gray}{ \textbf{Half Angle Formula}} \\ = -\int (1 - u^2) du \quad \textcolor{gray} {Let \space u = \cos(x). \space -du = \sin(x)dx} \\ = u - \frac{1}{3}u^3 + C \quad \textcolor{gray} {Integrate} \\ \int \sin^3(x) dx \space = \space \cos(x) - \frac{\cos^3(x)}{3} + C \quad \textcolor{gray} {Replace \space u \space with \space \cos(x)} $$