8.3.17-eg - A + C = 0-3 A + B-4 C = 3 2 A-B + 4 C =-1 ,...

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

View Full Document Right Arrow Icon
Example Evaluate Z 3 x - 1 ( x - 2)( x 2 - 3 x + 2) dx . 1. Factor the integrand and express as a sum of partial fractions. Notice that ( x - 2) is a repeated linear factor of the denominator: 3 x - 1 ( x - 2)( x 2 - 3 x + 2) = 3 x - 1 ( x - 2)( x - 1)( x - 2) = 3 x - 1 ( x - 2) 2 ( x - 1) so that partial fraction expansion is of the form: 3 x - 1 ( x - 2) 2 ( x - 1) = A x - 2 + B ( x - 2) 2 + C x - 1 . Multiplying through by the least common denominator ( x - 2) 2 ( x - 1) leads to the equation 3 x - 1 = A ( x - 2)( x - 1) + B ( x - 1) + C ( x - 2) 2 = Ax 2 - 3 Ax + 2 A + Bx - B + Cx 2 - 4 Cx + 4 C. Setting coefficients equal, we find A, B, C from the system of equations
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: A + C = 0-3 A + B-4 C = 3 2 A-B + 4 C =-1 , whose solution is A =-2 ,B = 5 ,C = 2. 2. Integrate. Rewrite the integral as Z 3 x-1 ( x-2)( x 2-3 x + 2) dx = Z-2 x-2 + 5 ( x-2) 2 + 2 x-1 dx. Using substitution in the second term ( u = x-2), we obtain Z 3 x-1 ( x-2)( x 2-3 x + 2) dx =-2 ln | x-2 | -5 x-2 + 2 ln | x-1 | + C = 2 ln ± ± ± ± x-1 x-2 ± ± ± ±-5 x-2 + C....
View Full Document

This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

Ask a homework question - tutors are online