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Question Number 96083 by john santu last updated on 29/May/20

$$\int \frac{{\mathrm{e}}^{\mathrm{x}}(1+\sin \mathrm{x})}{1+\cos \mathrm{x}}\mathrm{dx}$$

Answered by john santu last updated on 29/May/20

$$\int (\frac{e^x}{1+\cos x}+\frac{e^x\sin x}{1+\cos x})dx=$$$$\int [\frac{e^x}{2\cos {}^2\left(\frac{x}{2}\right)}+e^x\tan \left(\frac{x}{2}\right)]dx=$$$$\int [\frac{1}{2}{e}^x\sec {}^2\left(\frac{x}{2}\right)+e^x\tan \left(\frac{x}{2}\right)]dx=$$$$\int d\left(e^x\tan \left(\frac{x}{2}\right)\right)=e^x\tan \left(\frac{x}{2}\right)+c$$

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