# How do I evaluate this integral :$\int_0^\frac{\pi}{2} [\text{chi}(\cot^2x)+\text{shi}(\cot^2x)]\csc^2(x)e^{-\csc^2(x)}dx$?

I have tried to evaluate this integral :$$\int_0^\frac{\pi}{2} [\text{chi}(\cot^2x)+\text{shi}(\cot^2 x)]\csc^2(x)e^{-\csc^2(x)}dx$$ where, $\text{shi}(x)=\int_{0}^{x}\frac{\sinh t}{t}dt$ , $\text{chi}(x)=\gamma +\log(x)+\int_{0}^{x}\frac{\cosh(t)-1}{t} dt$

using integral by parts but I didn't succeed as I crossed this:
$$\int_0^\frac{\pi}{2}\csc^2(x)e^{-\csc^2(x)}dx$$ which gives : $\frac {\sqrt{\pi}}{2e}.$

I don't know how I can complete integration by parts since hasn't closed form .

Note : I guess this integral gives 0 (integration over closed path)

Thank you for any help !!!!!

• Is $\gamma$ Euler-Mascheroni constant? – Henricus V. Jan 1 '16 at 3:42
• according to the definition of CHi(x) i think it is Euler-Mascheroni constant – zeraoulia rafik Jan 1 '16 at 14:13
• seems that's gives 0 but proving it look hard integral – Salmahamizi Hamizi Jan 1 '16 at 15:23
• $$\int_0^{\pi/2} e^{-a^2 \csc^2 x} dx=\frac{\pi}{2} \text{erfc}(a)$$ – Yuriy S May 31 at 14:17
• I plotted the function and it appears to have a huge singularity at $x=0$, I'm not sure it's even integrable. – Yuriy S May 31 at 14:22