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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

image0.png

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:

    image1.png

    Make one more small rearrangement to place all the remaining x terms together:

    image2.png

    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,

    image3.png
  3. Substitute du/6 for x dx:

    image4.png

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

    image5.png
  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:

    image6.png
  5. To finish up, substitute 3x2 + 7 for u:

    image7.png

You can now check your integration by differentiating the result:

image8.png

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

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