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Question
\[\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{5/2}} dx\]
Solution
\[\int_\frac{\pi}{3}^\frac{\pi}{2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^\frac{5}{2}} d x\]
\[ = \int_\frac{\pi}{3}^\frac{\pi}{2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^\frac{5}{2}} \times \frac{\sqrt{1 - \cos x}}{\sqrt{1 - \cos x}} d x\]
\[ = \int_\frac{\pi}{3}^\frac{\pi}{2} \frac{\sin x}{\left( 1 - \cos x \right)^3}dx\]
\[ = - \frac{1}{2} \left[ \left( 1 - \cos x \right)^{- 2} \right]_\frac{\pi}{3}^\frac{\pi}{2} \]
\[ = - \frac{1}{2}\left[ 1 - 4 \right]\]
\[ = \frac{3}{2}\]
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