# show $\int_{-\pi}^{\pi} (4\arctan (e^x)-\pi)dx=0$ without calculating

Show $$\int_{-\pi}^{\pi} (4\arctan (e^x)-\pi)dx=0$$ without calculating.

I thought we should show that the integrand is odd, but I'm having trouble showing it.

If $f(x)=4\arctan (e^x)-\pi$ we need to show that $f(-x)=-f(x)$

so $f(-x)=4\arctan (e^{-x})-\pi$ and I'm not really see how to continue...

• $\arctan(x) + \arctan(\frac1{x}) = \frac{\pi}{2}$ – Nicholas Nov 12 '15 at 11:33
• @Nicholas exactly what I was missing, thanks! Do you want to make it an answer? – Stabilo Nov 12 '15 at 11:36

Note that $\arctan(x) + \arctan(\frac1{x}) = \frac{\pi}{2}$