My (physics) book gives the following approximation:
$\int_{-\pi/2}^{\pi/2} \sqrt{1-(1-a^2) \sin(k)^2} dk \approx 2 + (a_1 - b_1 \ln a^2) a^2 + O(a^2 \ln a^2)$
where a1 and b1 are "(unspecified) numerical constants." I've been looking for either a derivation of this, or the same approximation listed elsewhere and have gotten nowhere. Can someone help me along?
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