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Question Number 68481 by ajfour last updated on 11/Sep/19

$$I={\int }_0^1\sqrt{\frac{c-x^2}{x(1-x^2)}}dx(c>1)$$

Commented byMJS last updated on 11/Sep/19

$$\mathrm{I}\mathrm{tried}\mathrm{everything}\mathrm{I}\mathrm{know},\mathrm{it}\mathrm{seems}\mathrm{impossible}$$ $$\mathrm{to}\mathrm{solve}$$

Commented byKunal12588 last updated on 12/Sep/19

Commented byKunal12588 last updated on 12/Sep/19

Commented byMJS last updated on 12/Sep/19

$$\frac{2x\sqrt{1-x^2}\sqrt{\frac{c-x^2}{x-x^3}}}{\sqrt{1-\frac{x^2}{c}}}=2\sqrt{cx}$$ $$\Rightarrow {\left[2\sqrt{cx}{\mathrm{F}}_1(...)\right]}_0^1=2\sqrt{c}{\mathrm{F}}_1(\frac{1}{4};-\frac{1}{2},\frac{1}{2};\frac{5}{4};\frac{1}{c},1)$$

Commented byMJS last updated on 12/Sep/19

$$\mathrm{Sir}\mathrm{Kunal},\mathrm{can}\mathrm{you}\mathrm{calculate}\mathrm{it}\mathrm{now}?$$

Commented byMJS last updated on 12/Sep/19

$$\mathrm{I}\mathrm{found}\mathrm{this}:$$ $${\mathrm{F}}_1(a,b_1,b_2,c;x;y)=\frac{\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }(c-a)}\underset{0}{\stackrel{1}{\int }}{t}^{a-1}{(1-t)}^{c-a-1}{(1-xt)}^{-b_1}{(1-yt)}^{-b_2}dt$$ $$\mathrm{in}\mathrm{our}\mathrm{case}\mathrm{this}\mathrm{leads}\mathrm{to}$$ $$\frac{\sqrt{c}}{2}\underset{0}{\stackrel{1}{\int }}\frac{\sqrt{c-t}}{t^{3/4}\sqrt{c(1-t)}}dt$$ $$...\mathrm{but}\mathrm{is}\mathrm{this}\mathrm{easier}?$$

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