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Integration Techniques

Trigonometric Integrals

Techniques for integrating trigonometric functions

1. Powers of Sine and Cosine

Example 1.1: Odd Power of Sine

Find: ∫sin⁡3x dx\int \sin^3 x\,dx

Solution: Use sin⁡2x=1−cos⁡2x\sin^2 x = 1 - \cos^2 x

∫sin⁡3x dx=∫sin⁡x(1−cos⁡2x) dx\int \sin^3 x\,dx = \int \sin x(1-\cos^2 x)\,dx

Let u=cos⁡xu = \cos x, then du=−sin⁡x dxdu = -\sin x\,dx

=−∫(1−u2) du=−u+u33+C=−cos⁡x+cos⁡3x3+C= -\int (1-u^2)\,du = -u + \frac{u^3}{3} + C = -\cos x + \frac{\cos^3 x}{3} + C
Example 1.2: Even Powers

Find: ∫cos⁡2x dx\int \cos^2 x\,dx

Solution: Use double angle formula cos⁡2x=1+cos⁡2x2\cos^2 x = \frac{1+\cos 2x}{2}

∫cos⁡2x dx=∫1+cos⁡2x2 dx=x2+sin⁡2x4+C\int \cos^2 x\,dx = \int \frac{1+\cos 2x}{2}\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C

2. Products of Trig Functions

Example 2.1: Sin times Cos

Find: ∫sin⁡(3x)cos⁡(2x) dx\int \sin(3x)\cos(2x)\,dx

Solution: Use product-to-sum formula:

sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A\cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]
∫sin⁡(3x)cos⁡(2x) dx=12∫[sin⁡(5x)+sin⁡(x)] dx\int \sin(3x)\cos(2x)\,dx = \frac{1}{2}\int [\sin(5x) + \sin(x)]\,dx
=−110cos⁡(5x)−12cos⁡(x)+C= -\frac{1}{10}\cos(5x) - \frac{1}{2}\cos(x) + C

3. Tangent and Secant

Example 3.1: Powers of Tangent

Find: ∫tan⁡2x dx\int \tan^2 x\,dx

Solution: Use tan⁡2x=sec⁡2x−1\tan^2 x = \sec^2 x - 1

∫tan⁡2x dx=∫(sec⁡2x−1) dx=tan⁡x−x+C\int \tan^2 x\,dx = \int (\sec^2 x - 1)\,dx = \tan x - x + C
Example 3.2: Secant and Tangent

Find: ∫sec⁡xtan⁡x dx\int \sec x\tan x\,dx

Solution: Notice that ddx(sec⁡x)=sec⁡xtan⁡x\frac{d}{dx}(\sec x) = \sec x\tan x

∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec x\tan x\,dx = \sec x + C