7.4 - 7.4 Integration of Rational Functions by Partial...

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7.4: Integration of Rational Functions by Partial Fractions (Dated: October 22, 2011) Recall that a (real) polynomial is a function of the form P ( x ) = n X i = 0 a i x i = a n x n + a n - 1 x n - 1 +···+ a 1 x + a 0 , where n is a nonnegative integer and a i ’s are real numbers with a n 6= 0. The integer n is called the degree of the poly- nomial P , denoted deg( P ). This way any nonzero constant function is a polynomial of degree 0. It is convenient to con- sider the zero function 0 as a special polynomial whose de- gree is denoted deg(0) =-∞ . A rational function is a function of the form f ( x ) = P ( x ) Q ( x ) , where P and Q are polynomials. The rational function f is called improper if deg( P ) deg( Q ), otherwise it is called proper. In either case, there are polynomials S and Q such that f ( x ) = P ( x ) Q ( x ) = S ( x ) + R ( x ) Q ( x ) , where deg( R ) < deg( Q ). Note that P ( x ) = S ( x ) Q ( x ) + R ( x ). If f is proper, S = 0 and R = P . If f is improper, the polynomi- als S and R can be obtained by dividing Q into P by long di- vision until a remainder R is obtained with deg( R ) < deg( Q ). Let Q ( x ) = ( A 1 x 1 + B 1 ) m 1 ( A 2 x + B 2 ) m 2 ··· ( A k x + B k ) m k ( C 1 x 2 + D 1 x + E 1 ) n 1 ( C 2 x 2 + D 2 x + E 2 ) n 2 ··· ( C l x 2 + D l x + E l ) n l be the decomposition of Q into its linear factors and irre- ducible quadratic factors such that (1) no factor is repeated and (2) no factor is a constant multiple of another. Irre- ducibility here means the quadratic factors admit no real roots as polynomials. Then the partial fraction decompo- sition of R Q is the representation R ( x ) Q ( x ) = a 11 A 1 x + B 1 + a 12 ( A 1 x + B 1 ) 2 +···+ a 1 m 1 ( A 1 x + B 1 ) m 1 + a 21 A 2 x + B 2 + a 22 ( A 2 x + B 2 ) 2 +···+ a 2 m 2 ( A 2 x + B 2 ) m 2 +···+ a k 1 A k x + B k + a k 2 ( A k x + B k ) 2 +···+ a km k ( A k x + B k ) m k + c 11 x + d 11 C 1 x 2 + D 1 x + E 1 + c 12 x + d 12 ( C 1 x 2 + D 1 x + E 1 ) 2 +···+ c 1 n 1 x + d 1 n 1 ( C 1 x 2 + D 1 x + E 1 ) n 1 + c 21 x + d 21 C 2 x 2 + D 2 x + E 2 + c 22 x + d 22 ( C 2 x 2 + D 2 x + E 2 ) 2 +···+ c 2 n 2 x + d 2 n 2 ( C 2 x 2 + D 2 x + E 2 ) n 2 +···+ c l 1 x + d l 1 C l x 2 + D l x + E l + c l 2 x + d l 2 ( C l x 2 + D l x + E l ) 2 +···+ c ln l x + d ln l ( C l x 2 + D l x + E l ) n l . CASE 1: THE DENOMINATOR
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This note was uploaded on 11/11/2011 for the course MATH 152.01 taught by Professor Geline during the Fall '09 term at Ohio State.

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7.4 - 7.4 Integration of Rational Functions by Partial...

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