# Difficult Integration Problem

1. Sep 29, 2009

### EstimatedEyes

1. The problem statement, all variables and given/known data

Let $$f(x) = ax^2 + bx + c$$
$$a,b,c$$ are real numbers
$$f(0) = 1$$ and $$\int\frac{f(x)}{x^2(x+1)^3}dx$$ is a rational function.

Find $$f'(0)$$

2. Relevant equations

3. The attempt at a solution

$$f(0) = 1 = a(0)^2 + b(0) + c$$
$$c = 1$$
$$f'(x) = 2ax + b$$
$$f'(0) = 2a(0) + b = b$$
I tried to integrate $$\int\frac{f(x)}{x^2(x+1)^3}dx$$ and ended up doing some tedious partial fractions and ending up with logarithms (not rational?), so that's probably not the correct approach. I'm totally stuck at this point and any help would be greatly appreciated; Thanks!

2. Sep 29, 2009

### Staff: Mentor

$$\frac{ax^2 + bx + 1}{x^2(x + 1)^3}~=~\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 1} + \frac{D}{(x + 1)^2} + \frac{E}{(x + 1)^3}$$