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Section8_4_review

# Section8_4_review - Section 8.4 Integration of Rational...

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Section 8.4: Integration of Rational Functions by Partial Fractions This technique of integration allows us to express a rational function (which is a quotient of polynomials) as a sum of simpler fractions (called partial fractions), which are more easily integrated than the original rational function. In other words, we are faced with f x (29 dx , where f x ( 29 = P ( x ) Q ( x ) is a rational function. Let deg( P ) be the degree of the polynomial in the numerator and deg( Q ) be the degree of the polynomial in the denominator. Note that if deg( P ) < deg( Q ) , then f is a proper rational function; and if deg( P ) deg( Q ) , then f is an improper rational function. Procedure to integrate a rational function using partial fractions: Step 1. Determine if f is a proper or an improper rational function. (i) If f is proper, then f x ( 29 = P ( x ) Q ( x ) is the rational function that will be integrated using partial fractions. Go to Step 2. (ii) If f is improper, perform long division of polynomials to obtain f x (29 = Sx (29 + Rx (29 Qx (29 (where R is the remainder), and R x ( 29 Q x ( 29 is the rational function that will be integration using partial fractions. Go to Step 2. Step 2. Factor Qx ( 29 as far as possible. (The factored from of Qx ( 29 will contain linear and/or irreducible quadratic factors. An irreducible quadratic factor is a factor of the form ax 2 + bx + c for which b 2 - 4 ac < 0 .) Step 3. Perform the Partial Fraction Decomposition on the rational function.

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