Evaluate: $\int_{\frac{\pi}{2}}^{\frac{5\pi}{2}} \frac{e^{\cot^{-1}\cos(x)}}{e^{\cot^{-1}\sin(x)}+e^{\cot^{-1}\cos(x)}}dx$

Question

Evaluate: $\int_{\frac{\pi}{2}}^{\frac{5\pi}{2}} \frac{e^{\cot^{-1}\cos(x)}}{e^{\cot^{-1}\sin(x)}+e^{\cot^{-1}\cos(x)}}dx$

I used the property where we replace $x$ by $a+b-x$ where $a$ and $b$ are the lower and upper limits respectively (in this case $3\pi-x$) and simplified the result using properties of inverse trigonometric functions. I am stuck here and do not know how to proceed. Please help.

Answer 1

Put $f(x)=\exp\cot^{-1}\cos x$ and $g(x)=\exp\cot^{-1}\sin x$. $$\int_{\pi/2}^{5\pi/2}\frac{f(x)\,dx}{f(x)+g(x)}=\int_0^{2\pi}\frac{f(x)\,dx}{f(x)+g(x)}=\int_0^{2\pi}\frac{g(x)\,dx}{f(x)+g(x)}$$ (the first equality holds because of $2\pi$-periodicity of the integrand; the second is obtained by substituting $x=5\pi/2-y$ (and replacing $y$ by $x$) in the first (original) integral).

Now look at the sum of the last two integrals...