# How to Integrate a Function Multiplied by a Set of Nested Functions

Sometimes you need to integrate the product of a function (*x*) and a composition of functions (for example, the function 3*x*^{2} + 7 nested inside a square root function). If you were differentiating, you could use a combination of the Product Rule and the Chain Rule, but these options aren’t available for integration.

Given

here’s how you integrate, step by step, using variable substitution:

Declare a variable

*u*as follows and substitute it into the integral:Here, you assign a value to

*u**:*let*u*= 3*x*^{2}+ 7. Now substitute*u*into the integral:Make one more small rearrangement to place all the remaining

*x*terms together:This rearrangement makes clear that you still have to find a substitution for

*x dx**.*Now differentiate the function

*u*= 3*x*+ 7:^{2}This gives you the differential,

Substitute

*du*/6 for*x**dx:*You can move the fraction 1/6 outside the integral:

Now you have an integral that you know how to evaluate.

This example puts the square root in exponential form, to make sure that you see how to do this:

To finish up, substitute 3

*x*^{2}+ 7 for*u**:*

You can now check your integration by differentiating the result:

As if by magic, the derivative brings you back to the function you started with.