$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} d x=$

Question

Aicharya · Accepted Answer

Let $\begin{aligned} I & =\int_{\pi / 3}^{\pi / 2} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} d x \ & =\int_{\pi / 3}^{\pi / 2} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} 	imes \frac{\sqrt{1-\cos x}}{\sqrt{1-\cos x}} d x \ & =\int_{\pi / 3}^{\pi / 2} \frac{\sin x}{(1-\cos x)^3} d x\end{aligned}$ $\begin{array}{ll} & 	ext { Put } 1-\cos x=\mathrm{t} \ & \Rightarrow \sin x \mathrm{~d} x=\mathrm{dt} \ 	herefore \quad & \mathrm{I}=\int_{1 / 2}^1 \frac{\mathrm{dt}}{\mathrm{t}^3}=\left[\frac{\mathrm{t}^{-2}}{-2}ight]_{1 / 2}^1=\frac{3}{2}\end{array}$

$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} d x=$

Solution