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Evaluate the intergal:

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

Thank you

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7  
Hint: use symmetry –  Robert Israel Jun 19 '13 at 16:02
    
I don't believe that there is a closed form solution to this problem. –  SyntacticSugar Jun 19 '13 at 16:06
    
@StephenHerschkorn No. –  David Mitra Jun 19 '13 at 16:23
2  
@MuadDib42 There is, $\arctan(\pi^{x}) = \frac{\pi}{2} - \arctan(\pi^{-x})$. –  Shuhao Cao Jun 19 '13 at 16:27

2 Answers 2

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

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

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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?

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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

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