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Question Number 81801 by Power last updated on 15/Feb/20

Commented by mathmax by abdo last updated on 16/Feb/20

$$changement{x}^n={cos}^{2}tgivex={\left(cost\right)}^{\frac{2}{n}}\Rightarrow dx=-\frac{2}{n}sint{\left(cost\right)}^{\frac{2}{n}-1}$$$$\Rightarrow {\int }_0^1\frac{dx}{\left({}^n\sqrt{1-x^n}\right)}=\frac{2}{n}{\int }_0^{\frac{\pi }{2}}\frac{sint}{\left({}^n\sqrt{1-{cos}^{2}t}\right)}{\left(cost\right)}^{\frac{2}{n}-1}dt$$$$=\frac{2}{n}{\int }_0^{\frac{\pi }{2}}\frac{sint}{{sint}^{\frac{2}{n}}}{\left(cost\right)}^{\frac{2}{n}-1}dt=\frac{2}{n}{\int }_0^{\frac{\pi }{2}}{\left(sint\right)}^{1-\frac{2}{n}}{\left(cost\right)}^{\frac{2}{n}-1}dt$$$$wehavetheformula2{\int }_0^{\frac{\pi }{2}}{\left(cost\right)}^{2p-1}{\left(sint\right)}^{2q-1}=B(p,q)$$$$=\frac{\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(q\right)}{\mathrm{\Gamma }(p+q)}\Rightarrow {\int }_0^1\frac{dx}{({}^n\sqrt{1-x^n)}}=\frac{2}{n}{\int }_0^{\frac{\pi }{2}}{\left(sint\right)}^{2(1-\frac{1}{n})-1}{\left(cost\right)}^{\frac{2}{n}-1}dt$$$$=\frac{1}{n}B(1-\frac{1}{n},\frac{1}{n})=\frac{1}{n}\times \frac{\mathrm{\Gamma }(1-\frac{1}{n})\mathrm{\Gamma }\left(\frac{1}{n}\right)}{\mathrm{\Gamma }(1-\frac{1}{n}+\frac{1}{n})}=\frac{1}{n}\times \frac{\pi }{sin\left(\frac{\pi }{n}\right)}=\frac{\pi }{nsin\left(\frac{\pi }{n}\right)}$$

Answered by TANMAY PANACEA last updated on 15/Feb/20

$$x^n={sin}^2\theta \to x={\left(sin\theta \right)}^{\frac{2}{n}}$$$$dx=\frac{2}{n}{\left(sin\theta \right)}^{\frac{2}{n}-1}cos\theta d\theta$$$${\int }_0^{\frac{\pi }{2}}\frac{\frac{2}{n}{\left(sin\theta \right)}^{\frac{2}{n}-1}cos\theta d\theta }{{\left(cos\theta \right)}^{\frac{2}{n}}}$$$$\frac{1}{n}\times 2{\int }_0^{\frac{\pi }{2}}{\left(sin\theta \right)}^{\frac{2}{n}-1}.{\left(cos\theta \right)}^{1-\frac{2}{n}}d\theta$$$$formula$$$$2{\int }_0^{\frac{\pi }{2}}{\left(sin\theta \right)}^{2p-1}{\left(cos\theta \right)}^{2q-1}d\theta =\frac{\lceil \left(p\right)\lceil \left(q\right)}{\lceil (p+q)}$$$$2p-1=\frac{2}{n}-1\to p=\frac{1}{n}$$$$2q-1=1-\frac{2}{n}\to q=1-\frac{1}{n}$$$$p+q=1$$$$so=\frac{\lceil \left(\frac{1}{n}\right)\lceil (1-\frac{1}{n})}{\lceil \left(1\right)}=\frac{\pi }{sin\left(\frac{\pi }{n}\right)}$$$$answeris=\frac{1}{n}\times \frac{\pi }{sin\left(\frac{\pi }{n}\right)}$$$$$$$$formula\lceil \left(p\right)\lceil (1-p)=\frac{\pi }{sin\left(p\pi \right)}$$$$\mathit{p}\mathit{l}\mathit{s}\mathit{c}\mathit{h}\mathit{e}\mathit{c}\mathit{k}\mathit{m}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{k}\mathit{e}\mathit{i}\mathit{f}\mathit{a}\mathit{n}\mathit{y}$$

Commented by Power last updated on 15/Feb/20

$$\lceil -???$$

Commented by Tony Lin last updated on 15/Feb/20

$$\mathrm{\Gamma }\left(gammafunction\right)$$

Commented by TANMAY PANACEA last updated on 15/Feb/20

$$yessirthankyou...iforgottomention\lceil represent$$$$gammafunction$$

Commented by Power last updated on 15/Feb/20

$$\mathrm{thanks}$$

Commented by mathmax by abdo last updated on 16/Feb/20

$$youansweriscorrectsirtanmay$$

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