Question

Integrate the following function: $e^x\left(\frac{1+sinx}{1+cosx}\right)$

Answer 1

$\int e^x\left(\frac{1+sinx}{1+cosx}\right)dx=\int e^x\left(\frac{1+2sin\left(\frac{x}{2}\right)cos\left(\frac{x}{2}\right)}{1+2co{s}^2\left(\frac{x}{2}\right)}\right)dx=\int e^x\left[\right(\frac{1}{2}\left)se{c}^2\right(\frac{x}{2})+tan(\frac{x}{2}\left)\right]dxasf\left(x\right)=tan\left(\frac{x}{2}\right)then{f}^{\prime }\left(x\right)=se{c}^2\left(\frac{x}{2}\right)\cdot \left(\frac{1}{2}\right)dx\therefore integrationisoftheform\int e^x\left[f\right(x)+f^{\prime }(x\left)\right]dx=e^{x}f\left(x\right)\therefore I=e^{x}tan\left(\frac{x}{2}\right)+c$