Absolutely Atrocious

We are aware of the property

\int\limits_{-a}^{a} f(t) \mathrm{d}t = \left\{ \begin{array}{1 1} 0 & \quad \text{if \$f(t)\$ is odd}\\ 2\int\limits_0^{a} f(t) \mathrm{d}t & \quad \text{if \$f(t)\$ is even} \end{array} \right.

Hence we can split the given integrand into two functions. One is odd and one is even.

The integral with odd integrand reduces to $0$

The integral with even integrand reduces to

$\Rightarrow 2\int\limits_0^{1} \frac{1}{e^{|x|}} \mathrm{d}x$

$\Rightarrow 2\int\limits_0^{1} e^{-x} \mathrm{d}x$

$\Rightarrow -2.[e^{-x}]_0^1$

$\Rightarrow 2.(e^0 - e^{-1})$

$\Rightarrow 2(1 - e^{-1})$