How to evaluate the integral $\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx$

Question

Evaluate the intergal:

$$\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx.$$

Thank you

I don't believe that there is a closed form solution to this problem. — Yousuf Soliman, Jun 19 '13 at 16:06
@MuadDib42 There is, $\arctan(\pi^{x}) = \frac{\pi}{2} - \arctan(\pi^{-x})$. — Shuhao Cao, Jun 19 '13 at 16:27

score 4 · Answer 1 · answered Jun 19 '13 at 16:06

4

Hint: Use the fact that, for $y \gt 0$,

$$\arctan{y} + \arctan{\frac{1}{y}} = \frac{\pi}{2}$$

answered Jun 19 '13 at 16:06

Ron Gordon

138,521

score 1 · Answer 2 · answered Jun 19 '13 at 16:22

1

http://www.wolframalpha.com/ gives a nasty answer which involves the polylogarithmic function. I take this as evidence that the integral is not elementary. Are you sure you have not misstated this homework problem?

answered Jun 19 '13 at 16:22

Fred Rickey

691

You just use symmetry...... – Shuhao Cao Jun 19 '13 at 16:26
Shows a limitation of Wolfram. I get $\pi^2/2$. – Stephen Herschkorn Jun 19 '13 at 17:10

How to evaluate the integral $\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx$

2 Answers2