Question

If f(x) is a quadratic function such thatf(0)=1 and integral f(x) /x^2(x+1)^3 dx is a rational function

Answer 1

Let f(x)=ax2+bx+c. As f(0)=1, we have that f(x)=ax2+bx+1. Then

I=∫f(x)dxx2(x+1)3=∫a(x+1)3+bx(x+1)3+1x2(x+1)3dx=PartialFractions−1x+b−a−12(1+x)2+−2+b1+x+(b−3)log(x)−(b−3)log(1+x)+C.

As the function must be rational, we must have b=3, which leads f′(0)=2ax+b|x=0=3

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