Trigonometric Integrals

Powers of Sine and Cosine

$\int \sin^n(x) \cos^m(x) \, dx$

Case 1: n is odd ($n = 2k + 1$)

Save one $\sin(x)$ for $du$, convert rest using $\sin^2(x) = 1 - \cos^2(x)$:

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

Let $u = \cos(x)$, $du = -\sin(x) \, dx$

Case 2: m is odd ($m = 2k + 1$)

Save one $\cos(x)$ for $du$, convert rest using $\cos^2(x) = 1 - \sin^2(x)$:

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

Let $u = \sin(x)$, $du = \cos(x) \, dx$

Case 3: Both n and m are even

Use half-angle formulas:

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

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

$\sin(x)\cos(x) = \frac{\sin(2x)}{2}$

Powers of Tangent and Secant

$\int \tan^n(x) \sec^m(x) \, dx$

Case 1: m is even ($m = 2k$)

Save $\sec^2(x)$ for $du$, convert rest using $\sec^2(x) = 1 + \tan^2(x)$:

$\int \tan^n(x) \sec^{2k}(x) \, dx = \int \tan^n(x) (1+\tan^2 x)^{k-1} \sec^2(x) \, dx$

Let $u = \tan(x)$, $du = \sec^2(x) \, dx$

Case 2: n is odd ($n = 2k + 1$)

Save $\sec(x)\tan(x)$ for $du$, convert rest using $\tan^2(x) = \sec^2(x) - 1$:

$\int \tan^{2k+1}(x) \sec^m(x) \, dx = \int (\sec^2 x-1)^k \sec^{m-1}(x) \sec(x)\tan(x) \, dx$

Let $u = \sec(x)$, $du = \sec(x)\tan(x) \, dx$

Important Reduction Formulas

$\int \sin^n(x) \, dx = -\frac{\sin^{n-1}(x)\cos(x)}{n} + \frac{n-1}{n} \int \sin^{n-2}(x) \, dx$

$\int \cos^n(x) \, dx = \frac{\cos^{n-1}(x)\sin(x)}{n} + \frac{n-1}{n} \int \cos^{n-2}(x) \, dx$

$\int \tan^n(x) \, dx = \frac{\tan^{n-1}(x)}{n-1} - \int \tan^{n-2}(x) \, dx$

$\int \sec^n(x) \, dx = \frac{\sec^{n-2}(x)\tan(x)}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2}(x) \, dx$

Examples

Example 1: Odd power of sine

$\int \sin^3(x) \, dx = \int \sin^2(x) \cdot \sin(x) \, dx = \int (1-\cos^2 x)\sin(x) \, dx$

Let $u = \cos(x)$, $du = -\sin(x) \, dx$

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

Example 2: Both powers even

$\int \sin^2(x)\cos^2(x) \, dx = \int [\sin(x)\cos(x)]^2 \, dx$

$= \int \left[\frac{\sin(2x)}{2}\right]^2 dx = \frac{1}{4} \int \sin^2(2x) \, dx$

$= \frac{1}{4} \int \frac{1-\cos(4x)}{2} \, dx = \frac{1}{8}\left[x - \frac{\sin(4x)}{4}\right] + C$

Example 3: Even power of secant

$\int \sec^4(x) \, dx = \int \sec^2(x) \cdot \sec^2(x) \, dx = \int (1+\tan^2 x)\sec^2(x) \, dx$

Let $u = \tan(x)$, $du = \sec^2(x) \, dx$

$= \int (1+u^2) \, du = u + \frac{u^3}{3} + C = \tan(x) + \frac{\tan^3(x)}{3} + C$

Example 4: Odd power of tangent

$\int \tan^3(x) \, dx = \int \tan(x) \cdot \tan^2(x) \, dx = \int \tan(x)(\sec^2 x - 1) \, dx$

$= \int \tan(x)\sec^2(x) \, dx - \int \tan(x) \, dx$

For first integral, let $u = \tan(x)$:

$= \frac{\tan^2(x)}{2} - \ln|\sec(x)| + C$

Example 5: Mixed tan and sec

$\int \tan^3(x)\sec^3(x) \, dx$

Save $\sec(x)\tan(x)$, use $\tan^2 = \sec^2 - 1$:

$= \int \tan^2(x)\sec^2(x) \cdot \sec(x)\tan(x) \, dx$

$= \int (\sec^2 x - 1)\sec^2(x) \cdot \sec(x)\tan(x) \, dx$

Let $u = \sec(x)$, $du = \sec(x)\tan(x) \, dx$

$= \int (u^2 - 1)u^2 \, du = \int (u^4 - u^2) \, du$

$= \frac{u^5}{5} - \frac{u^3}{3} + C = \frac{\sec^5(x)}{5} - \frac{\sec^3(x)}{3} + C$