Trig Power Integrals

Level 1 - Math I (Physics) topic page in Integration.

Trigonometric Power Integrals

Integrals involving powers of trigonometric functions require different strategies based on whether the powers are even or odd.

For odd \(n\), separate one factor and convert the rest using \(\sin^2 x = 1 - \cos^2 x\) or \(\cos^2 x = 1 - \sin^2 x\).

Odd Sin

\[\int \sin^{2k+1} x \, dx = \int (\sin^2 x)^k \sin x \, dx = \int (1-\cos^2 x)^k \sin x \, dx\]

Example: \(\int \sin^3 x \, dx\)

Sin3

\[= \int (1-\cos^2 x) \sin x \, dx = \int (1-u^2)(-du) \quad (u = \cos x)\]

Result

\[= -\int (1-u^2) \, du = -u + \frac{u^3}{3} + C = -\cos x + \frac{\cos^3 x}{3} + C\]

Use half-angle identities:

Half-Angle

\[\sin^2 x = \frac{1-\cos 2x}{2}, \quad \cos^2 x = \frac{1+\cos 2x}{2}\]

Example: \(\int \cos^2 x \, dx\)

Cos2

\[\int \cos^2 x \, dx = \int \frac{1+\cos 2x}{2} \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C\]

For \(\tan^n x\), separate \(\tan^2 x = \sec^2 x - 1\):

Tan

\[\int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \tan x - x + C\]

Example: \(\int \tan^3 x \, dx\)

Tan3

\[\int \tan^3 x \, dx = \int \tan x(\sec^2 x - 1) \, dx = \int \tan x \sec^2 x \, dx - \int \tan x \, dx\]

Tan3 Result

\[= \frac{\tan^2 x}{2} - \ln|\sec x| + C\]

For \(\sec^n x\) with even \(n\), separate \(\sec^2 x\):

Sec

\[\int \sec^2 x \, dx = \tan x + C\]

Sec3

\[\int \sec^3 x \, dx = \int \sec x \cdot \sec^2 x \, dx = \frac{\sec x \tan x}{2} + \frac{1}{2}\ln|\sec x + \tan x| + C\]

Sin Reduction

\[\int \sin^n x \, dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2} x \, dx\]

Cos Reduction

\[\int \cos^n x \, dx = \frac{\cos^{n-1} x \sin x}{n} + \frac{n-1}{n} \int \cos^{n-2} x \, dx\]

These reduce the power by 2 each time, eventually reaching a solvable integral.