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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,


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:


    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:

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

  5. 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.

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