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0-1-1-x-4-dx-




Question Number 205506 by Lindemann last updated on 23/Mar/24
∫_0 ^1 (√(1−x^4 ))dx
$$\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx} \\ $$
Answered by sniper237 last updated on 23/Mar/24
=^(u=x^4 ) (1/4)∫_0 ^1 (1−u)^(1/2) u^((1/4)−1) du=(1/4)Beta((1/4);(3/2))
$$\overset{{u}={x}^{\mathrm{4}} } {=}\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{u}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} {u}^{\frac{\mathrm{1}}{\mathrm{4}}−\mathrm{1}} {du}=\frac{\mathrm{1}}{\mathrm{4}}{Beta}\left(\frac{\mathrm{1}}{\mathrm{4}};\frac{\mathrm{3}}{\mathrm{2}}\right) \\ $$

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