If f is quadratic function such that f(0)=1 and \int\frac{f(x)}{x^2(x+1)^3}dx

Fatima Traynor

Fatima Traynor

Answered question

2022-02-15

If f is quadratic function such that f(0)=1 and f(x)x2(x+1)3dx is a rational function, find the value of f′(0).
I already tried solving this question by using general quadratic equation ax2+bx+c and then using partial fraction method but it became very complicated.

Answer & Explanation

zakrwinpfo

zakrwinpfo

Beginner2022-02-16Added 4 answers

We have that b=f′(0) and c=f(0)=1. Moreover, by using partial fraction method we get 
ax2+bx+1x2(x+1)3=Ax+Bx2+Cx+1+D(x+1)2+E(x+1)3
A=0 and C=0 are the results of the integrals of those terms, which are logarithms, if the integral is a rational function, or a ratio of polynomials. Hence
ax2+bx+1=B(x+1)3+Dx2(x+1)+Ex2
that is 
(B+D)x3+(3B+D+Ea)x2+(3Bb)x+(B1)=0. Are you able to find b?

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