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 3x² + 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:

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

   Here, you assign a value to u: let u = 3x² + 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.

2. Now differentiate the function u = 3x² + 7:

   This gives you the differential,

   

3. Substitute du/6 for x dx:

   

   You can move the fraction 1/6 outside the integral:

   

4. 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:

   

5. To finish up, substitute 3x² + 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.