Partial Fractions Question: please help!!

Hi,
I know how to do this problem and get the answer in my book; but i was wondering why this method does not give the correct answer?

Here is the question:

∫ [(5x^2)+(3x)-(2)] / [(x^3)+(2x^2)] dx <--- (i know its hard to read but try writing it out please)

Ok so this is what i did:

step1: ∫ [ 5x-7]/(x^2) + 12/(x+2) dx <----- (got to that step by dividing x+2)

step2: ∫ 5/x - 7/(x^-2) + 12/[x+2] dx

step3: = 5 ln x + 7/x + 12 ln (x+2) + C


I know everything after step #1 is correct and I'm a 100% sure that

[ 5x-7]/(x^2) + 12/(x+2) = [(5x^2)+(3x)-(2)] / [(x^3)+(2x^2)]

...So Why Is This Method Invalid!?

Hey, 0zero0zero,

First of all, seriously, thanks for making your problem so legible. (You say it's hard to read, but actually, you included enough brackets and parentheses to make the problem clear. A lot of people don't do this, but it really helps.)

The trouble with your solution appears to be in Step 1, where you divide the numerator and denominator by (x + 2). Actually, the numerator doesn't have a factor of (x + 2) to be factored out.

The rational function we're integrating is:

f(x) = (5x^2 + 3x - 2) / (x^3 + 2x^2)

If we let f(x) = p(x) / q(x), so that:

p(x) = 5x^2 + 3x - 2
q(x) = x^3 + 2x^2

We can factor p(x) and q(x) as:

p(x) = (x + 1)(5x - 2)
q(x) = (x^2)(x + 2)

So, we can see the numerator p(x) and the denominator q(x) have no common factors. Instead of simplifying the fraction, we just have to convert it into the sum of partial fractions:

(5x^2 + 3x - 2) / (x^3 + 2x^2) = A/x^2 + B/x + C/(x + 2)
=>
5x^2 + 3x - 2 = A(x + 2) + Bx(x + 2) + Cx^2
=>
5x^2 + 3x - 2 = Ax + 2A + Bx^2 + 2Bx + Cx^2