u-Substitution

This is the most fundamental technique for simplifying integrals. It involves substituting a part of the integrand with a new variable, $ u $ , to make the integral easier to solve.

Steps:

1. Choose a suitable substitution, $ u = g(x) $ , where $ g(x) $ is a part of the integrand.
2. Find $ du = g’(x) dx $ .
3. Rewrite the integral in terms of $ u $ and $ du $ .
4. Integrate with respect to $ u $ .
5. Substitute back $ x $ for $ u $ in the result.

Example: $ \int x(x^2 + 1)^3 dx $

Let $ u = x^2 + 1 $ , then $ du = 2x dx $ , so $ x dx = \frac{1}{2} du $ .

The integral becomes: $ \int u^3 \frac{1}{2} du = \frac{1}{8}u^4 + C = \frac{1}{8}(x^2 + 1)^4 + C $

Techniques of Integration