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Question Number 190009 by Rupesh123 last updated on 26/Mar/23

Commented by Frix last updated on 26/Mar/23

$$\mathrm{Use}t=x^{\frac{\pi }{5}}\mathrm{to}\mathrm{get}\frac{5}{\pi }\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{dt}{t^{10}+1}\mathrm{and}\mathrm{now}‘‘{\mathrm{just}}^{″}$$$$\mathrm{decompose}\mathrm{etc}.$$

Commented by ARUNG_Brandon_MBU last updated on 27/Mar/23

$$\mathrm{which}\mathrm{is}\mathrm{equal}\mathrm{to}\frac{5}{\pi }\cdot \frac{1}{10}\beta (\frac{1}{10},\frac{9}{10})$$$$=\frac{1}{2\pi }\mathrm{\Gamma }\left(\frac{1}{10}\right)\mathrm{\Gamma }\left(\frac{9}{10}\right)=\frac{1}{2\pi }\cdot \frac{\pi }{\sin \left(\frac{\pi }{10}\right)}$$$$=\frac{1}{2}\mathrm{cosec}\left(\frac{\pi }{10}\right)$$

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