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I am working on the integral of the reciprocal of polynomial such that

$\int{\frac{x^{q-1}dx}{x^{p}-ax^{q}+1}}$

where $p$ and $q$ are coprime integers.

I tried to solve it by partial fraction decomposition as below

$\int{\frac{x^{q-1}dx}{x^{p}-ax^{q}+1}}\\ =\int{\prod_{j=1}^{p}\frac{x^{q-1}}{(x-k_j)}dx}\\ =\int{\sum_{j=1}^{p}\frac{A_j}{(x-k_j)}dx} $

where $k_{1,...,p}$ are solutions of the polynomial $x^{p}-ax^{q}+1=0$.

Is there any rules that solves the coefficients $A_j$? or Is the existence of the coefficients $A_j$ guaranteed?

Gonçalo
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  • What you did is very correct but the fact that $p$ and $q$ are coprime integer numbers does not change anything. – Claude Leibovici Nov 20 '23 at 11:44
  • Let $Q(x) = x^{q-1}$ and $P(x) = x^p - ax^q + 1$. If $P(x)$ and $P'(x)$ doesn't have common root, the roots $k_j$ of $P(x)$ are all simple and corresponding coefficient $A_j = \frac{Q(k_j)}{P'(k_j)} = \frac{1}{p k_j^{p-q} - aq}$. – achille hui Nov 20 '23 at 12:08

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