{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Steps for Integration by Partial Fractions

# Steps for Integration by Partial Fractions - 29 k x c there...

This preview shows page 1. Sign up to view the full content.

Steps for Integration by Partial Fractions Problem Statement : Evaluate ( 29 ( 29 p x dx q x , where p and q are polynomials with no common factors. So ( 29 1 1 0 n n n n p x a x a x a - - = + + + L , 0 n a , and ( 29 1 1 0 m m m m q x b x b x b - - = + + + L , 0 m b . Step 0 : If n m , divide the fraction out (using long division), obtaining a quotient Q and a remainder R for which ( 29 ( 29 ( 29 ( 29 ( 29 p x R x Q x q x q x = + , and the degree of R is smaller than the degree of q . Notice that Q is a polynomial and therefore easy to integrate. The problem is how to integrate ( 29 ( 29 R x q x . Partial Fractions is a technique that applies to this situation. Step 1 : Factor the denominator completely. For factors that are irreducible quadratics, complete the square so that they are of the form ( 29 2 2 k x c d - + . Of course, linear factors are of the form ( 29 k x c - . Step 2 : Decompose the integrand into the most general sum of fractions. For each linear factor in the denominator of the form
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ( 29 k x c-, there will be a sum of the form ( 29 ( 29 1 1 1 k k k k A A A x c x c x c--+ + +---L . For each irreducible quadratic factor in the denominator of the form ( 29 2 2 k x c d -+ , there is a sum of the form ( 29 ( 29 ( 29 1 1 1 1 1 2 2 2 2 2 2 k k k k k k B x C B x C B x C x c d x c d x c d---+ + + + + +-+ -+-+ L . The A s, B s, and C s are unknown constants. Step 3 : Clear the fractions in Step 2 and solve for the unknown constants. Step 4 : Using the results of Step 3, replace the integrand with the partial fractions decomposition and integrate each fraction. For linear factors , use substitution with u x c =-. For irreducible quadratics, use trig substitution with tan x c d θ- = ....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online