- #1
octowilli
- 10
- 0
Homework Statement
If f is a quadratic function such that f(0) = 1 and
[tex] \int \frac{f(x)}{x^2(x+1)^3}dx [/tex]
is a rational function, find the value of f '(0).
Homework Equations
The Attempt at a Solution
This question is presented in the context of learning about integration by partial fractions.
For the quadratic bit I have
[tex] f(x) = ax^2 + bx + c [/tex] [tex] f(0) = a(0)^2 + b(0) + c = c = 1 [/tex][tex] f'(x) = 2ax + b [/tex] [tex] f'(0) = 2a(0) + b = b [/tex]
So, I need to find the value of b? Plugging the integral into Wolfram|Alpha gives me
[tex] \int \frac{ax^2+bx+c}{x^2(x+1)^3}dx = -\frac{a-b+c}{2(x+1)^2} + \frac{b-2c}{x+1} + (b-3c)ln(x) - (b-3c)ln(x+1) - \frac{c}{x} + K [/tex]
I'll write out the work for the integral by hand when I submit the assignment. For now, I'm not sure how to relate f '(0) = b with any of this to find b, if it is b that I need to find. Thanks for commenting!