Question Number 127925 by mr W last updated on 03/Jan/21
$${find}\:{F}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\frac{\mathrm{1}+{a}^{\mathrm{2}} {t}^{\mathrm{2}} }{\mathrm{1}−{t}^{\mathrm{2}} }}\:{dt} \\ $$$$ \\ $$$${for}\:{background}\:{see}\:{Q}\mathrm{127811}. \\ $$
Answered by mindispower last updated on 03/Jan/21
$$=\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\frac{\mathrm{1}−\left(−\left({a}\right)^{\mathrm{2}} \right){t}^{\mathrm{2}} }{\mathrm{1}−{t}^{\mathrm{2}} }}{dt}={E}\left(\mathrm{1}\mid−{a}^{\mathrm{2}} \right) \\ $$$${E}\:{eleptic}\:{integral}\:{of}\:\mathrm{2}{nd}\:{kind} \\ $$$${E}\left({x}\mid{k}^{\mathrm{2}} \right)=\int_{\mathrm{0}} ^{{x}} \frac{\sqrt{\mathrm{1}−{k}^{\mathrm{2}} {t}^{\mathrm{2}} }}{\:\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}{dt} \\ $$
Commented by mr W last updated on 03/Jan/21
$${thanks}\:{sir}! \\ $$