Integration by substitution is a method for find antiderivatives of nested functions. It can loosely be thought of as using the chain rule "backwards".

An important aspect of substitution is choosing which term to substitute with . Ideally the term whose differential also occurs in the integrand, which will allow you to cancel it out.

When using integration by substitution, never substitution , because this will just rename without actually transforming the integral. Additionally, always re-order the equation so that it's easy to find as a factor, for example:

Another tip is to always simplify the equation first before applying integration by substitution.

Definition

Where is an integrable function, is a differentiable function, and is the antiderivative of

Examples

Indefinite Integrals

Example 1

Find

Example 2

Find

Example 3

Find

Definite Integral

Evaluate

Two Options for Solving