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{tex}\int \frac{\sqrt{1-\sin x} e^{-\pi / 2}}{1+\cos x} d x{/tex}
{tex}=\int \frac{\sqrt{\sin ^{2}\left(\frac{x}{2}\right)+\cos ^{2}\left(\frac{1}{2}\right)-2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)} e^{-x / 2}}{1+2 \cos ^{2}\left(\frac{x}{2}\right)-1} d x{/tex}
{tex}=\int \frac{\sqrt{\left(\sin \left(\frac{x}{2}\right)-\cos \left(\frac{x}{2}\right)\right)^{2}} e^{-x / 2}}{2 \cos ^{2}\left(\frac{x}{2}\right)} d x{/tex}
{tex}\int \frac{\sqrt{1-\sin x} e^{-x / 2}}{1+\cos x} d x{/tex}
{tex}=\int\left(\frac{\sin \left(\frac{x}{2}\right)}{2 \cos ^{2}\left(\frac{\pi}{2}\right)}-\frac{\cos \left(\frac{x}{2}\right)}{2 \cos ^{2}\left(\frac{\pi}{2}\right)}\right) e^{-x / 2} d x{/tex}
{tex}=\int\left[\frac{1}{2} \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) e^{-x / 2}-\frac{1}{2} \sec \left(\frac{x}{2}\right) e^{-x / 2}\right] d x{/tex}
{tex}=\frac{1}{2} \int \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) e^{-x / 2} d x-\frac{1}{2} \int \sec \left(\frac{x}{2}\right) e^{-x / 2} d x{/tex}
{tex}\left.=\frac{1}{2} \int \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) e^{-x / 2} d x-\frac{1}{2}\left[\sec \left(\frac{x}{2}\right)\right] e^{-x / 2} d x-\int \frac{d}{d x}\left(\sec \left(\frac{x}{2}\right)\right)\left(\int e^{-x / 2} d x\right) d x\right] \quad(U\text { sing integration by parts) }{/tex}
{tex}=\frac{1}{2} \int \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) e^{-x / 2} d x-\frac{1}{2}\left[\sec \left(\frac{x}{2}\right)\left(\frac{e^{-x / 2}}{-1 / 2}\right)-\int \frac{1}{2} \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right)\left(\frac{e^{-x / 2}}{-1 / 2}\right) d x\right]{/tex}
{tex}=\frac{1}{2} \int \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) e^{-x / 2} d x-\frac{1}{2} \sec \left(\frac{x}{2}\right)\left(\frac{e^{-x / 2}}{-1 / 2}\right)+\frac{1}{2} \int \frac{1}{2} \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right)\left(\frac{e^{-x / 2}}{-1 / 2}\right) d x{/tex}
{tex}=\frac{1}{2} \int \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) e^{-x / 2} d x+\sec \left(\frac{x}{2}\right) e^{-x / 2}-\frac{1}{2} \int \tan \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) e^{-x / 2} d x{/tex}
{tex}=\sec \left(\frac{x}{2}\right) e^{-x / 2}+C{/tex}
{tex}\Rightarrow \int \frac{\sqrt{1+\sin x}}{\cos x} e^{-x / 2} d x=\sec \left(\frac{x}{2}\right) e^{-x / 2}+C{/tex}
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