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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, sin3 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,

image0.png

follow these steps:

  1. 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:

    image1.png

    Notice that the expression cos x dx still remains and needs to be expressed in terms of u.

  2. Differentiate the function u = sin x.

    This gives you the differential du = cos x dx.

  3. Substitute du for cos x dx in the integral:

    image2.png
  4. Now you have an expression that you can integrate:

    image3.png
  5. Substitute sin x for u:

    image4.png

Now check this answer by differentiating with the Chain Rule:

image5.png

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

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