8.4 - Calculus II-Stewart Dr. Berg Spring 2010 8.4...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 8.4 8.4 Integration by Partial Fractions We can integrate rational functions by writing them as sums of simpler fractions. Example A Since 1 x 1 + 1 x 2 + 1 = x 2 + 1 ( ) + x 1 ( ) x 1 ( ) x 2 + 1 ( ) = x 2 + x x 3 x 2 + x 1 , then x 2 + x x 3 x 2 + x 1 dx = 1 x 1 + 1 x 2 + 1 dx = ln x 1 + arctan x + C . Basic Idea Given a rational function P ( x ) Q ( x ) , we begin by completely factoring the denominator Q ( x ) and then write P ( x ) Q ( x ) as a sum of rational functions (partial fractions) whose denominators are (powers of) the factors of Q ( x ) . We encounter four cases: linear factors of multiplicity one, linear factors of higher multiplicity, irreducible quadratic factors of multiplicity one, and irreducible quadratic factors of higher multiplicity. The first step is to use long division to rewrite the rational function as a polynomial plus a proper rational function (the degree of the numerator is smaller than
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8.4 - Calculus II-Stewart Dr. Berg Spring 2010 8.4...

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