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Question Number 125421 by bemath last updated on 11/Dec/20

$$\underset{0}{\stackrel{100}{\int }}\frac{dx}{\sqrt{x(100-x)}}?$$

Answered by liberty last updated on 11/Dec/20

$$I=\underset{0}{\stackrel{100}{\int }}\frac{dx}{\sqrt{x}\sqrt{100-x}};let\sqrt{x}=10\sin t$$$$dx=100.\left(2\sin t\cos t\right)dt$$$$I=\underset{0}{\stackrel{\pi /2}{\int }}\frac{200\sin t\cos t}{10\sin t\left(10\cos t\right)}dt=2\underset{0}{\stackrel{\pi /2}{\int }}dt$$$$I=2\times \frac{\pi }{2}=\pi$$

Answered by Dwaipayan Shikari last updated on 11/Dec/20

$${\int }_0^{100}\frac{dx}{\sqrt{x(100-x)}}u=100x$$$$={\int }_0^1\frac{du}{\sqrt{u}\left(\sqrt{1-u}\right)}=\beta (\frac{1}{2},\frac{1}{2})={\mathrm{\Gamma }}^2\left(\frac{1}{2}\right)=\pi$$

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