$\int \frac{\sqrt{1 - \sin x}}{1 + \cos x} e^{- x / 2} d x$

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Solution

$Let I = \int (\frac{\sqrt{1 - \sin x}}{1 + \cos x}) e^{\frac{- x}{2}} d x = \int (\frac{\sqrt{\cos^{2} \frac{x}{2} + \sin^{2} \frac{x}{2} - 2 \sin \frac{x}{2} \cos \frac{x}{2}}}{2 \cos^{2} \frac{x}{2}}) e^{\frac{- x}{2}} d x = \int \frac{\sqrt{{(\cos \frac{x}{2} - \sin \frac{x}{2})}^{2}}}{2 \cos^{2} \frac{x}{2}} e^{\frac{- x}{2}} d x = \int (\frac{\sin \frac{x}{2} - \cos \frac{x}{2}}{2 \cos^{2} \frac{x}{2}}) e^{\frac{- x}{2}} d x = \int [\frac{1}{2} \sec \frac{x}{2} \tan \frac{x}{2} - \frac{1}{2} \sec (\frac{x}{2})] e^{\frac{- x}{2}} d x = \frac{1}{2} \int (\sec \frac{x}{2} \tan \frac{x}{2} - \sec \frac{x}{2}) e^{\frac{- x}{2}} d x let e^{\frac{- x}{2}} \sec (\frac{x}{2}) = t Diff both sides w . r . t x e^{\frac{- x}{2}} \frac{\sec (\frac{x}{2}) \tan (\frac{x}{2})}{2} + \sec (\frac{x}{2}) \times e^{\frac{- x}{2}} \times \frac{- 1}{2} = \frac{d t}{d x} \Rightarrow {\frac{e}{2}}^{\frac{- x}{2}} [\sec \frac{x}{2} \tan \frac{x}{2} - \sec \frac{x}{2}] d x = d t ∴ \frac{1}{2} \int (\sec \frac{x}{2} \tan \frac{x}{2} - \sec \frac{x}{2}) e^{\frac{- x}{2}} d x = \int d t \Rightarrow I = \int t + C = e^{\frac{- x}{2}} \sec (\frac{x}{2}) + C$

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