I need to calculate the next integral $$\int_{0}^{\infty}{dx \over \sqrt[n]{1+x^n}}. $$
I tried doing $u=x^n$ to then develop it as a beta function but did not achieve anything.
Thanks for your help!
2026-03-25 05:50:49.1774417849
Calculate $\int_{0}^{\infty}{dx \over \sqrt[n]{1+x^n}} $
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Use $x=\tan^{2/n}t$ so the integral is $$\frac{2}{n}\int_0^2\tan^{2/n-1}t\sec^{2-2/n}tdt=\frac{2}{n}\int_0^2\sin^{2/n-1}t\cos^{-1}tdt=\frac{1}{n}\operatorname{B}(\frac{1}{n},\,0).$$This diverges, as can be seen by writing the result in terms of Gamma functions. The divergence is unsurprising, since for large $x$ the integrand is asymptotically $1/x$.