Solving f'(0): Integral of x^-2 (x-5)^-5 f(x) dx

[iamtheman]

Anyone have any ideas how to solve this? (Find f'(0)

Let f(x) be a quadratic function such that f(0)=6 and

integral of (f(x) / x^2 (x-5)^5) dx

f(x)
---------- dx
x^2 (x-5)^5


is a rational function.
Determine the value of f'(0)

Thanks!

 

[himanshu121]

zero 0

 

[phoenixthoth]

Anyone have any ideas how to solve this?

i don't see how zero answers this question.

i could see zero as being the solution, if that were correct, but not HOW to solve it.

the key fact is that the antiderivative of a rational function of the form k x^n is itself rational iff n!=-1. so use partial fractions and write f(x) as a general quadratic. then integrate it. since you know the integral is a rational function, that means NO LOGS can be in the integral when you're done. that will put a constraint on the coefficients in f.

f(0)=6 puts another constraint on the coefficients of f (well, one coefficient).

then write f'(0) in terms of the coefficients of f. using the constraints found earlier, this should hopefully give you f'(0) and I'm guessing you won't even be able to determine all coefficients of f yet you will be able to tell what f'(0) is.

i get f'(0)=-6 but i might have done it wrong.

another tip: you don't need to do the whole integral once you see what will make logs disappear.