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Question Number 127925 by mr W last updated on 03/Jan/21

find F(a)=∫_0 ^1 (√((1+a^2 t^2 )/(1−t^2 ))) dt    for background see Q127811.

$${find}\:{F}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\frac{\mathrm{1}+{a}^{\mathrm{2}} {t}^{\mathrm{2}} }{\mathrm{1}−{t}^{\mathrm{2}} }}\:{dt} \\ $$$$ \\ $$$${for}\:{background}\:{see}\:{Q}\mathrm{127811}. \\ $$

Answered by mindispower last updated on 03/Jan/21

=∫_0 ^1 (√((1−(−(a)^2 )t^2 )/(1−t^2 )))dt=E(1∣−a^2 )  E eleptic integral of 2nd kind  E(x∣k^2 )=∫_0 ^x ((√(1−k^2 t^2 ))/( (√(1−t^2 ))))dt

$$=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\frac{\mathrm{1}−\left(−\left({a}\right)^{\mathrm{2}} \right){t}^{\mathrm{2}} }{\mathrm{1}−{t}^{\mathrm{2}} }}{dt}={E}\left(\mathrm{1}\mid−{a}^{\mathrm{2}} \right) \\ $$$${E}\:{eleptic}\:{integral}\:{of}\:\mathrm{2}{nd}\:{kind} \\ $$$${E}\left({x}\mid{k}^{\mathrm{2}} \right)=\int_{\mathrm{0}} ^{{x}} \frac{\sqrt{\mathrm{1}−{k}^{\mathrm{2}} {t}^{\mathrm{2}} }}{\:\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}{dt} \\ $$

Commented by mr W last updated on 03/Jan/21

thanks sir!

$${thanks}\:{sir}! \\ $$

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