Evaluate double integral $\int_{-1}^1 \int_{-4|x|}^{|x|} e^{x+y} \, dy \, dx$ involving absolute values

Precisely! We have $$\begin{align}\int_{-1}^1 e^{x+|x|} - e^{x - 4|x|} \, \mathrm{d}x &= \int_{-1}^0 e^{x \color{red}{-}x}-e^{x\color{red}{+}4x} \, \mathrm{d}x + \int_0^1 e^{x\color{green}{+}x} - e^{x\color{green}{-}4x} \, \mathrm{d}x \\ & = \int_{-1}^0 1 - e^{5x} \, \mathrm{d}x+\int_0^1 e^{2x} -e^{-3x} \, \mathrm{d}x \\ & = \bigg[x-\frac{1}{5}e^{5x}\bigg]_{-1}^0 + \bigg[\frac{1}{2}e^{2x} + \frac{1}{3}e^{-3x}\bigg]_0^1 \\ & = \frac{4}{5}-\frac{e^{-5}}{5} + \frac{e^2}{2} + \frac{e^{-3}}{3} - \frac{5}{6}\end{align}$$

Indeed, the easiest thing to do is split the integral at zero: on $[-1,0)$, $\lvert x \rvert = -x$ so one has $e^{0}-e^{5x}$, and on $(0,1]$, $\lvert x \rvert = x$, so the integrand is $ e^{2x}-e^{-3x} $.