How Do You Solve Integrals Using Partial Fractions?

[zeppelinpage4]

Homework Statement

I basically need to find the integral of (-5x^3-1x^2+4)/(x^4+2x^3)

Homework Equations

Just the basic rules associated with partial fractions when re-writing a polynomial with constants.

The Attempt at a Solution

Since the degree of the denominator is 4 and the degree of the numerator is 3 I went straight to factoring out the denominator.
I ended up getting
(x^4+2x^3)=x^3(x+2)

From this point I went forward with the basic steps (I think I made the mistake here, I'm not sure how to re-write the form for (x^3), so I broke it into (x^2) & (x)).

[(-5x^3-1x^2+4)/(x^4+2x^3)]= [(Ax+B)/(x^2)]+[C/x]+[D/(x+2)]

When I multiply both sides by (x^4+2x^3) (this is the denominator on the left side)

I end up with

(-5x^3-1x^2+4)=x^3(A+C+D) + x^2(2A+B+2C) + x(2B)

Normally i'd solve for the constants A, B, C and D to fit the polynomial on the left side BUT I don't know how to account for the "+ 4" at the end of (-5x^3-1x^2+4).

I know
A+C+D=-5
2A+B+2C=-1
2B=0

?=4

I'm very lost at how to approach this and any advice or help that could give me some clue would be very much appreciated. It's been a tough week with midterms and my calc teacher hasn't been helping. =P

 

[epenguin]

This bit (x^4+2x^3)=x^3(x+2) is fine.

Personally I would look at the last two terms of the numerator, ask if they look anything familar. I find using that it falls out quite simply to something you can integrate. When you have got that, knowing the answer you will be able to check through your general method, which is what you have to use when you are not so lucky, and clear up where you went wrong maybe.

But there is virtue in simplicity, and I wouldn't present an unnecessarily laborious version.