# Thread: Tricky integral problem.. I've completely reached a dead end :(

1. ## Tricky integral problem.. I've completely reached a dead end :(

"If f is a quadratic function such that f(0) = 1 and ∫(f(x)/(x^2(x+1)^3) dx is a rational function (no ln and arctan terms) find the value of f'(0)."

So what I've done so far is is make f(x)/(x^2(x+1)^3) = (Ax+B)/x ...... (Kx^3 + Lx +M)/(x+1)^3. I have 5 terms total like those. So then I multiplied everything by the denominator under f(x). I then multiplied it all out and organized them like this, for ex: (A +C + F + H +K)x^5, etc. so I made the x^5, x^4, and x^3 coefficients all = 0, and I found that E = 1 (I just went down the alphabet as I had made the numerators previously). I'm not sure where to go from here bc I don't know what coefficient from the left side of the equation to equate to my x^2 or x coefficients.

^ my method descriptions may be kind of confusing, you can ignore them if you like and start fresh.

I've tried some other methods as well, but nothing's working
Any help will be appreciated!! thanks

2. Originally Posted by kelseyk23
"If f is a quadratic function such that f(0) = 1 and ∫(f(x)/(x^2(x+1)^3) dx is a rational function (no ln and arctan terms) find the value of f'(0)."

So what I've done so far is is make f(x)/(x^2(x+1)^3) = (Ax+B)/x ...... (Kx^3 + Lx +M)/(x+1)^3.
wrong! ignoring the exponents, the terms in the denominator are $x$ and $x+1,$ which are of degree 1. so, partial fractions gives you: $\frac{f(x)}{x^2(x+1)^3}=\frac{\alpha}{x}+\frac{\be ta}{x^2} + \frac{\gamma}{x+1} + \frac{\delta}{(x+1)^2} + \frac{\mu}{(x+1)^3}.$

more hints:

Spoiler:

so $I=\int \frac{ax^2+bx+c}{x^2(x+1)^3} \ dx$ is rational if and only if $\frac{ax^2+bx+c}{x^2(x+1)^3}=\frac{A}{x^2} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3}.$ we're given that $c=f(0)=1.$ now show that $f'(0)=b=3.$

3. Originally Posted by kelseyk23
"If f is a quadratic function such that f(0) = 1 and ∫(f(x)/(x^2(x+1)^3) dx is a rational function (no ln and arctan terms) find the value of f'(0)."

So what I've done so far is is make f(x)/(x^2(x+1)^3) = (Ax+B)/x ...... (Kx^3 + Lx +M)/(x+1)^3. I have 5 terms total like those. So then I multiplied everything by the denominator under f(x). I then multiplied it all out and organized them like this, for ex: (A +C + F + H +K)x^5, etc. so I made the x^5, x^4, and x^3 coefficients all = 0, and I found that E = 1 (I just went down the alphabet as I had made the numerators previously). I'm not sure where to go from here bc I don't know what coefficient from the left side of the equation to equate to my x^2 or x coefficients.

^ my method descriptions may be kind of confusing, you can ignore them if you like and start fresh.

I've tried some other methods as well, but nothing's working
Any help will be appreciated!! thanks
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