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Question Number 205590 by Lindemann last updated on 25/Mar/24

$$J={\int }_0^1\sqrt{1-x^4}dx$$

Commented by lepuissantcedricjunior last updated on 25/Mar/24

$$\mathit{J}={\int }_0^1\sqrt{1-{\mathit{x}}^4}\mathit{d}\mathit{x}\mathit{p}\mathit{o}\mathit{s}\mathit{o}\mathit{n}\mathit{s}{\mathit{x}}^4=\mathit{t}=>\mathit{x}=\sqrt[4]{\mathit{t}}$$$$\iff \mathit{d}\mathit{x}=\frac{\mathit{d}\mathit{t}}{4{\mathit{t}}^{\frac{3}{4}}}=$$$$\mathit{q}\mathit{d}:\{\begin{array}{l}\mathit{x}\to 0\\ \mathit{x}\to 1\end{array}=>\{\begin{array}{l}\mathit{t}\to 0\\ \mathit{t}\to 1\end{array}$$$$\mathit{J}=\frac{1}{4}{\int }_0^1{\mathit{t}}^{-\frac{3}{4}}{(1-\mathit{t})}^{\frac{1}{2}}\mathit{d}\mathit{t}=\frac{1}{4}{\int }_0^1{\mathit{t}}^{\frac{1}{4}-1}{(1-\mathit{t})}^{\frac{3}{2}-1}\mathit{d}\mathit{t}$$$$\mathit{J}=\frac{1}{4}\left(\frac{\mathbf{\Gamma }\left(\frac{1}{4}\right)\times \mathbf{\Gamma }\left(\frac{3}{2}\right)}{\mathbf{\Gamma }\left(\frac{7}{4}\right)}\right)=\frac{1}{4}\left(\frac{\frac{1}{2}\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathbf{\Gamma }\left(\frac{1}{4}\right)}{\frac{3}{4}\mathbf{\Gamma }\left(\frac{3}{4}\right)}\right)$$$$=\frac{\mathrm{\Gamma }\left(\frac{1}{4}\right)\sqrt{\mathit{\pi }}}{6\mathbf{\Gamma }\left(\frac{3}{4}\right)}$$$$\mathit{J}=\frac{\mathrm{\Gamma }\left(\frac{1}{4}\right)\sqrt{\mathit{\pi }}}{6\mathrm{\Gamma }\left(\frac{3}{4}\right)}$$$$..............lepuissant\mathit{D}r...................$$

Commented by mr W last updated on 26/Mar/24

$$pleasepostyouranswertoaquestion$$$$as‘‘{answer}^{″},notas‘‘{comment}^{″}!thanks!$$

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