So, set b – 3 = 0 and –b + 3 = 0 and solve for b.

[EstimatedEyes]

Homework Statement

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 $$f'(0)$$

Homework Equations

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!

 

[Mark44]

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

After multiplying both sides by x²(x + 1)³ and grouping the terms by powers of x, I got these values for the constants A through E.
A = b - 3
B = 1
C = -b + 3
D = b - 4
E = a - 3b + 7
(Caveat: I didn't double-check my work, so there might be an error or two.)

If my numbers are correct, there is one value of b that gives a coefficient of 0 for the 1/x term and the 1/(x + 1) term (the ones that produce log terms when integrated). The other terms do not produce log terms when integrated.