Singular integral equation calculus of variation kernel

Question

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.

Answer 1

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.

Answer 2

\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.