The integral Ι = ∫ e^ x ((1 + sin x)/(1 + cos x))dx = e^ x f (x) +...

The given integral $Ι = ∫ e^{x}\left(\dfrac{1 + 2sin\dfrac{x}{2}cos\dfrac{x}{2}}{2cos^{2}\dfrac{x}{2}}\right)dx$
$= ∫ e^{x}\left(\dfrac{1}{2}sec^{2}\left(\dfrac{x}{2}\right) + \tan \left(\dfrac{x}{2}\right)\right)dx$
$= e^{x} ⋅ \text{tan} \left(\dfrac{x}{2}\right) +$C \left\{∵ ∫ e^{x}\left(f\left(x\right) + f\text{'}\left(x\right)\right)\left(dx = e^{x}f \left(x\right)+ C\right)\right\}
Thus, $f\left(x\right)$ = \tan \left(\dfrac{x}{2}\right) whose range is \left(- \infty, \infty\right)