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Question Number 97465 by hosein_golmakani last updated on 08/Jun/20

$${\int }_0^{\propto }{e}^{-x^4}dx=\frac{1}{4}$$$$pleaseproveit$$

Commented by PRITHWISH SEN 2 last updated on 08/Jun/20

$$\mathrm{let}{\mathrm{x}}^4=\mathrm{z}$$$$\mathrm{dx}=\frac{1}{4}{\mathrm{z}}^{-\frac{3}{4}}\mathrm{dz}$$$$\frac{1}{4}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{z}}{\mathrm{z}}^{(\frac{1}{4}-1)}\mathrm{dz}=\frac{1}{4}\mathrm{\Gamma }\left(\frac{1}{4}\right)$$

Answered by smridha last updated on 08/Jun/20

$$\mathit{l}\mathit{e}\mathit{t}{\mathit{x}}^2=\mathit{t}$$$$\mathit{s}\mathit{o}\mathit{I}=\frac{1}{2}{\int }_0^{\mathrm{\infty }}{\mathit{e}}^{-{\mathit{t}}^2}{\mathit{t}}^{-\frac{1}{2}}\mathit{d}\mathit{t}$$$$=\frac{1}{4}\left[2{\int }_0^{\mathrm{\infty }}{\mathit{e}}^{-{\mathit{t}}^2}{\mathit{t}}^{2\frac{1}{4}-1}\mathit{d}\mathit{t}\right]$$$$=\frac{1}{4}\mathbf{\Gamma }\left(\frac{1}{4}\right)$$$$\mathit{I}\mathit{t}\mathit{h}\mathit{i}\mathit{n}\mathit{k}\mathit{A}\mathit{n}\mathit{s}\mathit{i}\mathit{s}\mathit{w}\mathit{r}\mathit{o}\mathit{n}\mathit{g}..$$

Answered by mathmax by abdo last updated on 08/Jun/20

$$\mathrm{error}\mathrm{in}\mathrm{the}\mathrm{question}...!$$

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