partial fractions is a helpful trick to solve integrals of
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7.4: Partial Fractions
Partial fractions is a helpful trick to solve integrals of rational functions. Consider the following integral:
By our current techniques, we have no way to do this. However, notice the following algebra fact:
So, we can finish the integral in this manner:
So, how do we find this "decomposition"? We follow a method called partial fractions.
Partial Fractions: If the degree of the numerator is less than the degree of the denominator, we can decompose any rational function into a sum of rational functions whose denominators are the factors of the original denominator.
While this looks complicated, we'll take it step by step. First, we must ensure that the degree of the numerator is lower. If not, we'll use long division.
Example: Evaluate
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Once the degree of the numerator is lower, we factor the denominator. We then have 4 cases.
Case 1: The denominator is a product of distinct linear factors.
Example: Evaluate
Example: Evaluate (cont)
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*Example: Evaluate
(be able to factor 2x3 + 5x2 + 3x in the future)
*Example: Evaluate (cont)
(be able to factor 2x3 + 5x2 + 3x in the future)
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Case 2: One factor is a repeated linear factor.
Example: Evaluate
Example: Evaluate (cont)
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Example: Evaluate (cont)
*Example: Evaluate
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Case 3: One factor is an irreducible quadratic factor.
In this case, write your other factors, and the quadratic part is
Example: Evaluate
Example: Evaluate (cont)
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*Example: Evaluate
Case 4: One factor is a repeated irreducible quadratic factor.
Example: Evaluate
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Example: Evaluate (cont)
Example: Evaluate (cont)
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Case 4: One factor is a repeated irreducible quadratic factor.
Example: Evaluate
(Alternate method of partial fractions)