Identity involving an improper integral (with geometric application) – mathoverflow.net

Is it (for some reason) true that $\lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt=\frac{\pi}{2}$? Numerical evidence (from Mathematica): when $c=1/5$, the integral is $\...

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