Question
$\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^{\frac{5}{2}}}$
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Answer
Let $\text{I}=\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^{\frac{5}{2}}}$
$\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^2\sqrt{1-\cos\text{x}}}\text{dx}$
$\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{1}{(1-\cos\text{x})^2}\text{dx}=\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{1}{(\sin^2\text{x})}\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\cos\text{ec}^2\text{x dx}=\big[-\cot\text{x}\big]^{\frac{\pi}{2}}_{\frac{\pi}{3}}$
$=-\Big[\cot\frac{\pi}{2}-\cot\frac{\pi}{2}\Big]$ $=-\Big[0-\frac{1}{\sqrt{3}}\Big]=+\frac{1}{\sqrt{3}}$
$\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^2\sqrt{1-\cos\text{x}}}\text{dx}$
$\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{1}{(1-\cos\text{x})^2}\text{dx}=\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{1}{(\sin^2\text{x})}\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\cos\text{ec}^2\text{x dx}=\big[-\cot\text{x}\big]^{\frac{\pi}{2}}_{\frac{\pi}{3}}$
$=-\Big[\cot\frac{\pi}{2}-\cot\frac{\pi}{2}\Big]$ $=-\Big[0-\frac{1}{\sqrt{3}}\Big]=+\frac{1}{\sqrt{3}}$
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