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Pol. whose antider. of the recipr. of their assoc. f are frac. of rational pol.

Source: Romania National Olympiad 2015, grade xii, problem 4

August 23, 2019
algebrapolynomialfunctioncalculusintegration

Problem Statement

Find all non-constant polynoms f∈Q[X] f\in\mathbb{Q} [X] that don't have any real roots in the interval [0,1] [0,1] and for which there exists a function ξ:[0,1]⟶Q[X]×Q[X],ξ(x):=(gx,hx) \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) such that hx(x)≠0 h_x(x)\neq 0 and ∫0xdtf(t)=gx(x)hx(x), \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , for all x∈[0,1]. x\in [0,1] .
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