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How to solve this integral? $$\int_{T-s}^{T}\frac{1}{t^n(T-t)^n}\mathbb{d}t, \qquad (n \in \mathbb{N}) $$ This integral type singular integral of order $n$. Please help me to solve or give some reference books.

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  • $\begingroup$ Do you have a bound for $s$? $\endgroup$ – Henricus V. Jan 2 '17 at 5:56
  • $\begingroup$ yes, for $s\in (T/2,T)$ and $n\in\mathbb{N}$ is finite. $\endgroup$ – kamalakkannan Jan 4 '17 at 6:20
  • $\begingroup$ @kamalakkannan See my edited answer $\endgroup$ – polfosol Jan 4 '17 at 7:10
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\begin{align} \int \frac{1}{x^{n}(T-x)^{n}} dx &= \frac{1}{T^{n}} \int \frac{1}{x^{n}(1-x/T)^{n}} dx \\ &= T^{1-2n} \int y^{-n} (1-y)^{-n} dy \\ &= T^{1-2n} \mathrm{B}_{y}(1-n,1-n) \\ &= T^{1-2n} \mathrm{B}_{x/T}(1-n,1-n) \\ &= \frac{1}{1-n} \frac{x^{1-n}}{T^{n}} {}_{2}\mathrm{F}_{1}(1-n,n;2-n;x/T) \end{align}

We have used the incomplete Beta function and Gauss's hypergeometric function.

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Similar to a process I have discussed in here, the integral can be converted to $$I=\left(\frac 2T\right)^{2n-1}\int_0^{\theta_0}(\sin x)^{1-2n}dx$$ where $\theta_0=\arccos(1-\frac{2s}T)$. With your assumptions, the integral is divergent.

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