How to Integrate Compositions of Functions
How to Integrate a Function Multiplied by a Set of Nested Functions
Substituting with Expressions of the Form f(x) Multiplied by g(x)

Using the Product Rule to Integrate the Product of Two Functions

The Product Rule enables you to integrate the product of two functions. For example, through a series of mathematical somersaults, you can turn the following equation into a formula that’s useful for integrating.

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This derivation doesn’t have any truly difficult steps, but the notation along the way is mind-deadening, so don’t worry if you have trouble following it. Knowing how to derive the formula for integration by parts is less important than knowing when and how to use it.

The first step is simple: Just rearrange the two products on the right side of the equation:

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Next, rearrange the terms of the equation:

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Now integrate both sides of this equation:

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Use the Sum Rule to split the integral on the right in two:

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The first of the two integrals on the right undoes the differentiation:

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This is the formula for integration by parts. But because it’s so hairy looking, the following substitution is used to simplify it:

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Here’s the friendlier version of the same formula, which you should memorize:

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Understanding What Makes a Function Integrable
How to Antidifferentiate Any Polynomial Using the Sum, Constant Multiple, and Power Rules
Setting Up Partial Fractions When You Have Repeated Linear Factors
Substituting with Expressions of the Form f(x) Multiplied by h(g(x))
How to Integrate Rational Expressions Using the Sum, Constant Multiple, and Power Rules
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