Prove:∫_0 ^1 (√((((√(K^2 +36K′^2 ))+6K^′ )/(K^2 +36K^(′2) )) ))(dk/( (√k)(1−k^2 )^(2/3) ))=(√π)((√2)−(√((4−2(√2))/3)))


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Question Number 217683 by MrGaster last updated on 18/Mar/25

$Prove : \int_{0}^{1} \sqrt{\frac{\sqrt{K^{2} + 36 K^{' 2}} + 6 K^{^{'}}}{K^{2} + 36 K^{^{'} 2}}} \frac{d k}{\sqrt{k} {(1 - k^{2})}^{\frac{2}{3}}} = \sqrt{π} (\sqrt{2} - \sqrt{\frac{4 - 2 \sqrt{2}}{3}})$
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