# Singular integral equation calculus of variation kernel

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.

• Do you have a bound for $s$? – Henricus V. Jan 2 '17 at 5:56
• yes, for $s\in (T/2,T)$ and $n\in\mathbb{N}$ is finite. – kamalakkannan Jan 4 '17 at 6:20
• @kamalakkannan See my edited answer – polfosol Jan 4 '17 at 7:10

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.