# Substituting with Expressions of the Form *f*(*x*) Multiplied by *g*(*x*)

When *g*'(*x*) = *f*(*x*), you can use the substitution *u* = *g*(*x*) to integrate expressions of the form *f*(*x*) multiplied by *g*(*x*). Variable substitution helps to fill the gaps left by the absence of a Product Rule and a Chain Rule for integration.

Some products of functions yield quite well to variable substitution. Look for expressions of the form *f*(*x*) multiplied by *g*(*x*) where

You know how to integrate

*g*(*x*).The function

*f*(*x*) is the derivative of*g*(*x*).

For example:

The main thing to notice here is that the derivative of tan *x* is sec^{2} *x**.* This is a great opportunity to use variable substitution:

Declare

*u*and substitute it into the integral:Differentiate as planned:

Perform another substitution:

This integration couldn’t be much easier:

Substitute back tan

*x*for*u:*