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duynhan1

$$\begin{aligned}I &= \int \frac{1-\sin x}{1 + \cos x } .e^{-x} \text{dx} \\ & = \int \frac{e^{-x}}{2 \cos^2{\frac{x}{2}}} \text{dx} + \int \tan{\frac{x}{2}} \mathrm{d(e^{-x})} \\ &= \int \frac{e^{-x}}{2 \cos^2{\frac{x}{2}}} \text{dx} + \tan {\frac{x}{2}} . e^{-x} - \int \frac{e^{-x}}{2 \cos^2{\frac{x}{2}}} \text{dx} + C \\ & = \tan { \frac{x}{2} } .e^{-x} + C \end{aligned}$$
 
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