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Evaluating this integral for different values of a constant

Is there a solution to the definite integral, $$\int\limits_{0}^{\infty} \frac{1}{x^{\frac{1}{n}}}\frac{1}{1+x^2}\mathrm{d}x$$ where, $n \in \mathbb{N}$

HINT : substitute, $x = \tan\theta$

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marked as duplicate by joriki, martini, Asaf Karagila, Did, t.b. Aug 11 '12 at 9:11

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Are you giving us a hint ? –  Belgi Aug 11 '12 at 7:12
    
I wonder if there's something special about $n\in\Bbb N$ and $x=\tan\theta$ that makes this question seek a different type of answer than the one given in the duplicate question. –  anon Aug 11 '12 at 7:25
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1 Answer

Maple says that $$\int\limits_0^{\infty}x^{-1/n}\dfrac 1{x^2+1}dx=\dfrac 12\pi\sec\left(\dfrac{\pi}{2n}\right)$$

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