I-0-1-c-x-2-x-1-x-2-dx-c-gt-1-

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

I = \int_{0}^{1} \sqrt{\frac{c - x^{2}}{x (1 - x^{2})}} d x (c > 1)

Commented by MJS last updated on 11/Sep/19

I tried everything I know, it seems impossible

to solve

Commented by Kunal12588 last updated on 12/Sep/19

Commented by Kunal12588 last updated on 12/Sep/19

Commented by MJS last updated on 12/Sep/19

\frac{2 x \sqrt{1 - x^{2}} \sqrt{\frac{c - x^{2}}{x - x^{3}}}}{\sqrt{1 - \frac{x^{2}}{c}}} = 2 \sqrt{c x}

\Rightarrow {[2 \sqrt{c x} F_{1} (\dots)]}_{0}^{1} = 2 \sqrt{c} F_{1} (\frac{1}{4}; - \frac{1}{2}, \frac{1}{2}; \frac{5}{4}; \frac{1}{c}, 1)

Commented by MJS last updated on 12/Sep/19

Sir Kunal, can you calculate it now ?

Commented by MJS last updated on 12/Sep/19

I found this :

F_{1} (a, b_{1}, b_{2}, c; x; y) = \frac{Γ (c)}{Γ (a) Γ (c - a)} \underset{0}{\int^{1}} t^{a - 1} {(1 - t)}^{c - a - 1} {(1 - x t)}^{- b_{1}} {(1 - y t)}^{- b_{2}} d t

in our case this leads to

\frac{\sqrt{c}}{2} \underset{0}{\int^{1}} \frac{\sqrt{c - t}}{t^{3 / 4} \sqrt{c (1 - t)}} d t

\dots but is this easier ?

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