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, $n \in \mathbb{N}$

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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, $x = \tan\theta$

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 07:25

score 1 · Answer 1 · answered Aug 11 '12 at 08:47

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

answered Aug 11 '12 at 08:47

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, $n \in \mathbb{N}$

1 Answers1