The problem of fraction decomposition it's easy to solve by using the cover up method, when there are no repeated linear factors in the denominator . Nevertheless it could turn into a hard work if these factors are raised to a high power, where the cover up method doesn't work. This technique shows how to calculate these coefficients without solving large systems of equations with a clever rearrangement of the numerator.

shortly from the Espil's theorem, we can derive the generalized Dirichlet integral for any natural value when the hole integrand is raised to the n-th power.

The problem of integration technique over integrands of the form f(t)/t^n, can be solved by differentiation(n times) by using Leibniz's rule to get rid of t^n, that leads to integrate back (n times) to end the game which it's harder than the original problem.This work focuses on the derivation of the formula (Espil's Theorem) which is a perfect tool to avoid that hard task. It allows to change the difficult n iterated integrals into a more outstanding easier problem which consists of n-1 ...