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Integration of Rational Functions by Partial Fractions Motivation: Evaluating the integral x 2 x5 dx would be much easier if we knew x2 ahead of time that x5 2 1 x x 2 x 1 x 2 2 P( x) as a sum of simpler Q( x) fractions provided the degree of P is less than the degree of Q. In general, it is possible to express a rational function f ( x) A Preliminary Example. Integrate the function x3 x x 1 dx . Some Algebraic Background Any polynomial Q can be factored as a product of linear factors (of the form ax + b) and irreducible quadratic factors (of the form ax2 + bx + c where the discriminant b2 – 4ac < 0). For example, factor x4 – 16. Case 1: The denominator Q(x) is a product of distinct linear factors. That is, Q( x) (a1 x b1 )( a2 x b2 ) (ak x bk ) where no factor is repeated. Then the partial fraction theorem states that there exist constants A1, A2, …, Ak such that Ak A1 A2 P( x) Q( x) a1 x b1 a2 x b2 ak x bk Examples x 2 2x 1 2 x 3 3x 2 2 x dx x 2 dx a2 Case 2: Q(x) is a product of linear factors, some of which are repeated. If, for example, the linear factor (a1x + b1) is repeated r times, i.e. (a1x + b1)r appears in factorization of Q(x), then instead of the single term A1/(a1x + b1) in the expansion, we have A1 A2 Ar 2 a1 x b1 (a1 x b1 ) (a1 x b1 ) r Example. Write the form of the partial fraction decomposition of x3 x 1 . x 2 ( x 1) 3 x 4 2x 2 4x 1 Example. Evaluate the integral 3 dx . x x2 x 1 Case 3: Q(x) contains irreducible quadratic factors, none of which is repeated. If Q(x) has a factor ax2 + bx + c (irreducible), then the partial fraction expansion will have a term of the form Ax B ax bx c 2 where A and B are constants to be determined. Example. Write the form of the p. f. decomposition of x . ( x 2)( x 1)( x 2 4) A useful integration rule follows: x 2 dx 1 x arctan C 2 a a a 2 Examples 2x 2 x 4 x 3 4 x dx 4 x 2 3x 2 4 x 2 4 x 3 dx Case 4: Q(x) contains a repeated irreducible quadratic factor. If Q(x) has the factor (ax2 + bx + c)r, then the following expression occurs in the partial fraction decomposition: A1 x B1 A2 x B2 Ar x Br 2 2 2 ax bx c (ax bx c) (ax 2 bx c) r Example. Write the form of the partial fraction decomposition of the function x3 x2 1 . x( x 1)( x 2 x 1)( x 2 1) 3 Examples 1 x 2x 2 x3 x( x 2 1) 2 dx x4 dx x