Question

$\underset{\mathrm{\pi }/3}{\overset{\mathrm{\pi }/2}{\int }}\frac{\sqrt{1+\cos x}}{{\left(1-\cos x\right)}^{5/2}}dx$

Answer 1

$$\begin{gathered} {\int }_{\frac{\mathrm{\pi }}{3}}^{\frac{\mathrm{\pi }}{2}}\frac{\sqrt{1+\cos x}}{{\left(1-\cos x\right)}^{{\displaystyle \frac{5}{2}}}}dx \\ ={\int }_{\frac{\mathrm{\pi }}{3}}^{\frac{\mathrm{\pi }}{2}}\frac{\sqrt{1+\cos x}}{{\left(1-\cos x\right)}^{{\displaystyle \frac{5}{2}}}}\times \frac{\sqrt{1-\cos x}}{\sqrt{1-\cos x}}dx \\ ={\int }_{\frac{\mathrm{\pi }}{3}}^{\frac{\mathrm{\pi }}{2}}\frac{\sin x}{{\left(1-\cos x\right)}^3}dx \\ =-\frac{1}{2}{\left[{\left(1-\cos x\right)}^{-2}\right]}_{\frac{\mathrm{\pi }}{3}}^{\frac{\mathrm{\pi }}{2}} \\ =-\frac{1}{2}\left[1-4\right] \\ =\frac{3}{2} \end{gathered}$$