# Finding the Integral of a Product of Two Functions

Sometimes the function that you’re trying to integrate is the product of two functions — for example, sin^{3} *x* and cos *x.* This would be simple to differentiate with the Product Rule, but integration doesn’t have a Product Rule. Fortunately, variable substitution comes to the rescue.

Given the example,

follow these steps:

Declare a variable as follows and substitute it into the integral:

Let

*u*= sin*x*You can substitute this variable into the expression that you want to integrate as follows:

Notice that the expression

*cos x dx*still remains and needs to be expressed in terms of*u**.*Differentiate the function

*u*= sin*x**.*This gives you the differential

*du*= cos x*dx**.*Substitute

*du*for*cos x dx*in the integral:Now you have an expression that you can integrate:

Substitute sin

*x*for*u:*

Now check this answer by differentiating with the Chain Rule:

This derivative matches the original function, so the integration is correct.