Monstrous Integral

Put $x = \tan z$ $\Rightarrow \text{dx} = \sec^2z \text{dz}$ , and transform the integral to

$\displaystyle \int_{0}^{\frac{\pi}{4}} e^z \sec z \tan^2 z \text{dz}$ ,

First we find the indefinite integral,

$I' = \displaystyle \int e^z \sec z tan^2 z \text{dz}$

= $\displaystyle \int e^z \sec^3 z \text{dz} - \int e^z \sec z \text{dz}$

Apply by parts on $\displaystyle \int e^z \sec z \text{dz}$ to get:

$I'$ = $\displaystyle \int e^z \sec^3 z \text{dz} - e^z \sec z + \int e^z \sec z \tan z \text{dz}$

Again apply by parts on $\displaystyle \int e^z \sec z \tan z \text{dz}$ to get :

$I' = \displaystyle \int e^z \sec^3 z \text{dz} - e^z \sec z + e^z \sec z \tan z - \int e^z(\sec^3 z - \tan^2 z \sec z) \text{dz}$

$\Rightarrow I' = e^z \sec z \tan z - e^z \sec z - I'$

$\Rightarrow I' = \frac{1}{2} \bigg(e^z(\sec z \tan z - \sec z)\bigg)$

Now just apply the limits $0$ to $\frac{\pi}{4}$ to get $\boxed {4I=2}$