∫ e^ x [(2 + 2x)/(1 + 2x)]dx =

Let,& $I = ∫ e^{x}\left[\dfrac{2 + \sin 2x}{1 + \cos 2x}\right]dx$

We know that~$\sin 2x = 2\sin x \cos x \text{ and } \cos 2x = 2\cos^{2}x - 1$

Using the above formulas we can write

$I = ∫ e^{x}\left(\dfrac{2 + 2\sin x \cos x}{2\cos^{2}x}\right)dx$

$I = ∫ e^{x}\left(\dfrac{2\left(1 + \sin x \cos x\right)}{2\cos^{2}x}\right)dx$

$I = ∫ e^{x}\left(\dfrac{1}{\cos^{2}x} + \dfrac{\sin x \cos x}{\cos^{2}x}\right)dx$
$I = ∫ e^{x} ( \tan x + \sec^{2} x )dx$

We know that~$∫ e^{x}\left(\text{f}\left(\text{x}\right) + f^{'}\left(\text{x}\right)\right)\text{dx} = e^{x} \cdot \text{f} + \text{C}$

Using the above formula we can write

$I = e^{x} \tan x + C$