Integration by substitution

The idea

Substitution turns an integral into a simpler one in a new variable $u$. It works well when the derivative of an inner function also appears: in $\int 2x(x^2 + 1)^4\,dx$, use $u = x^2 + 1$ because $\frac{du}{dx} = 2x$.

1. Choose $u$ (often the inside of a bracket or a power).
2. Replace $dx$ using $\frac{du}{dx}$, so no $x$ is left.
3. Integrate in $u$. Change the limits to $u$-values, or substitute back.

Watch out Change the limits when you change the variable, or go back to $x$ before using them.