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How to Integrate Rational Expressions Using the Sum, Constant Multiple, and Power Rules

In many cases, you can untangle hairy rational expressions and integrate them using the anti-differentiation rules plus the Sum Rule, Constant Multiple Rule, and Power Rule.

The Sum Rule for integration tells you that integrating long expressions term by term is okay. Here it is formally:

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The Constant Multiple Rule tells you that you can move a constant outside of a derivative before you integrate. Here it is expressed in symbols:

image1.png

The Power Rule for integration allows you to integrate any real power of x (except –1). Here’s the Power Rule expressed formally:

image2.png

Here’s an integral that looks like it may be difficult:

image3.png

You can split the function into several fractions, but without the Product Rule or Quotient Rule, you’re then stuck. Instead, expand the numerator and put the denominator in exponential form:

image4.png

Next, split the expression into five terms:

image5.png

Then use the Sum Rule to separate the integral into five separate integrals and the Constant Multiple Rule to move the coefficient outside the integral in each case:

image6.png

Now you can integrate each term separately using the Power Rule:

image7.png
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