Mathematics

Mathematics

Problem

∫1−cos⁡xcos⁡α−cos⁡x dx\int \sqrt{\frac{1 - \cos x}{\cos \alpha - \cos x}}\,dx where 0<α<x<π0 < \alpha < x < \pi, is equal to (a) 2ln⁡(cos⁡α2−cos⁡x2)+c2\ln\left(\cos \frac{\alpha}{2} - \cos \frac{x}{2}\right) + c (b) 2ln⁡(cos⁡α2−cos⁡x2)+c\sqrt{2}\ln\left(\cos \frac{\alpha}{2} - \cos \frac{x}{2}\right) + c (c) 22ln⁡(cos⁡α2−cos⁡x2)+c2\sqrt{2}\ln\left(\cos \frac{\alpha}{2} - \cos \frac{x}{2}\right) + c (d) −2sin⁡−1(cos⁡x2cos⁡α2)+c-2\sin^{-1}\left(\frac{\cos \frac{x}{2}}{\cos \frac{\alpha}{2}}\right) + c (a) 2ln⁡(cos⁡α2−cos⁡x2)+c2\ln\left(\cos \frac{\alpha}{2} - \cos \frac{x}{2}\right) + c (b) 2ln⁡(cos⁡α2−cos⁡x2)+c\sqrt{2}\ln\left(\cos \frac{\alpha}{2} - \cos \frac{x}{2}\right) + c (c) 22ln⁡(cos⁡α2−cos⁡x2)+c2\sqrt{2}\ln\left(\cos \frac{\alpha}{2} - \cos \frac{x}{2}\right) + c (d) −2sin⁡−1(cos⁡x2cos⁡α2)+c-2\sin^{-1}\left(\frac{\cos \frac{x}{2}}{\cos \frac{\alpha}{2}}\right) + c

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