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Question Number 80515 by M±th+et£s last updated on 03/Feb/20

$$\int \frac{dx}{{(1+x^{\varphi })}^{\varphi }}$$

Commented by john santu last updated on 04/Feb/20

$$t=1+x^{\varphi }\Rightarrow x={(t-1)}^{\frac{1}{\varphi }}$$$$I={\int }_0^{\mathrm{\infty }}\frac{1}{t^{\varphi }}.\frac{1}{\varphi }{(t-1)}^{\frac{1}{\varphi }-1}dt$$$$=\frac{1}{\varphi }{\int }_0^{\mathrm{\infty }}{t}^{-\varphi +{\varphi }^{-1}}{(1-t^{-1})}^{\frac{1}{\varphi }-1}dt$$$$=\frac{1}{\varphi }.\frac{\mathrm{\Gamma }(\varphi -\frac{1}{\varphi })\mathrm{\Gamma }\left(\frac{1}{\varphi }\right)}{\mathrm{\Gamma }\left(\varphi \right)}$$$$=\frac{1}{\varphi }.\frac{\mathrm{\Gamma }\left({\varphi }^{-1}\right)}{{\varphi }^{-1}\mathrm{\Gamma }\left({\varphi }^{-1)}\right)}=1$$$$where\varphi =\frac{\sqrt{5}-1}{2}$$

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