How to Integrate by Using Partial Fractions when the Denominator Contains Only Linear Factors

You can use the partial fractions method to integrate rational functions (Recall that a rational function is one polynomial divided by another.) The basic idea behind the partial fraction approach is “unadding” a fraction:

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Before using the partial fractions technique, you have to check that your integrand is a “proper” fraction — that’s one where the degree of the numerator is less than the degree of the denominator. If the integrand is “improper,” like

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you have to first do long polynomial division to transform the improper fraction into a sum of a polynomial (which sometimes will just be a number) and a proper fraction. Here’s the division for this improper fraction (without explanation). Basically, it works just like regular long division.

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With regular division, if you divide 4 into 23, you get a quotient of 5 and a remainder of 3, which tells you that

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The result of the above polynomial division tells you the same thing.

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The first integral is just 2x. You would then do the second integral with the partial fractions method.

Here’s how the method works, but let’s tackle a less complicated integral than the one immediately above; this will make the technique easier to follow.

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  1. Factor the denominator.

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  2. Break up the fraction on the right into a sum of fractions, where each factor of the denominator in Step 1 becomes the denominator of a separate fraction. Then put unknowns in the numerator of each fraction.

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  3. Multiply both sides of this equation by the denominator of the left side.

    This is algebra I, so you can’t possibly want to see the steps. Right?

    5 = A(x + 3) + B(x – 2)

  4. Take the roots of the linear factors and plug them — one at a time — into x in the equation from Step 3, and solve for the unknowns.

    If x = 2

    5 = A(2 + 3) + B(2 – 2)

    5 = 5A

    A = 1

    Or if x = –3

    5 = A(–3 + 3) + B(–3 – 2)

    5 = –5B

    B = –1

  5. Plug these results into the A and B in the equation from Step 2.

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  6. Split up the original integral into the partial fractions from Step 5 and you’re home free.

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