Integration Techniques

Special Integration Techniques

Advanced methods for challenging integrals

1. Weierstrass Substitution

Example 1.1: Universal Trig Substitution

For rational functions of sine and cosine, use $t = \tan(x/2)$

Formulas:

\sin x = \frac{2t}{1+t^2}, \quad \cos x = \frac{1-t^2}{1+t^2}, \quad dx = \frac{2}{1+t^2}\,dt

This converts any trig rational function to an algebraic rational function!

Example 2.1: Power Reduction

For $I_n = \int \sin^n x\,dx$ , we can derive:

I_n = -\frac{1}{n}\sin^{n-1}x\cos x + \frac{n-1}{n}I_{n-2}

This allows us to reduce the power step by step.

Example 3.1: Adding and Subtracting

Find: $\int \frac{x^2}{(x\sin x + \cos x)^2}\,dx$

Hint: Add and subtract $1$ in the numerator:

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

This can split into simpler parts.