6.9 Integrating Using Substitution

Substitution is used when an integrand contains a function and its derivative together in a way that matches the reverse of the chain rule. If differentiation of a composite function looks like f(g(x))g'(x) , then integration can often be simplified by letting ${u=g(x)}$. This changes the integral into a simpler form in terms of u, which is often much easier to antidifferentiate. The goal is to recognize when part of the integrand can be treated as an inside function and when the remaining part matches, or is a constant multiple of, its derivative.

To use substitution, you choose a new variable ${u}$, compute ${du}$, rewrite the integral, integrate with respect to u, and then substitute back in terms of x. This works best when the integrand has a clear inner expression raised to a power, inside a square root, inside an exponential, or inside a trig function, and when its derivative is present nearby. You should think of substitution as the integration version of the chain rule: if the derivative of the inside is there, the integral often becomes much simpler after the change of variable.