I was reading a paper where the following integral appears:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]I = \int_0 ^{\pi}dt\sqrt{k^2 + \sin^2t}[/tex]

In the limit [itex]k^2 \ll 1[/itex] the authors present the following approximation

[tex]I \approx 2 + \frac{k^2}{2}\left(\ln{\frac{1}{k^2}} + 4\ln 2 + 1\right).[/tex]

I'm trying to reproduce this result but with no luck. Any idea how it should go? I've plotted both expressions as a function of k and they indeed agree for small k.

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# Help with an integral

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