Expert: Paul Klarreich - 11/7/2008

Question
Evaluate the Integral:  Integral sign (5x^2+3x-2)/(x^3+2x^2) dx

Answer
Questioner:   Alfred Phiri
Category:  Calculus
Private:  No
 
Subject:  Integration of Rational Functions by Partial Fractions
Question:  Evaluate the Integral:  Integral sign (5x^2+3x-2)/(x^3+2x^2) dx
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Hi, Alfred,

It would be a good idea for you to send me what you already tried to do.  Otherwise, I have to assume you just need to get started.

You get started by:

1. Factoring the bottom:

5x^2+3x-2
--------- =
x^3+2x^2

5x^2 + 3x - 2
--------------
x^2(x + 2)

Writing the (hypothetical) reduction:

5x^2 + 3x - 2    Ax + B      C
-------------- = ------- + ------
x^2(x + 2)         x^2     x + 2

Clearing fractions (multiply by your LCD)

5x^2 + 3x - 2  =  (Ax + B)(x + 2) + Cx^2

5x^2 + 3x - 2  =  Ax^2 + 2Ax + Bx + 2B + Cx^2

Now equate coefficients:

x^2   :  5 = A + C
x     :  3 = 2A + B
const : -2 = 2B

Solve for A,B,C.  Then rewrite your reduction and integrate each term.