How do I integrate the following integral?
$$\int_{-\infty}^{\infty} \! e^{-|x-y|} \, \mathrm{d}x.$$
My approach is the divide the calculation into two parts: $$\int_{0}^{\infty} \! e^{-|x-y|} \, \mathrm{d}x + \int_{-\infty}^{0} \! e^{-|(-x)-y|} \, \mathrm{d}x.$$ and the my result is: $$e^y + e^{-y}$$
I don't think my answer is correct and I couldn't find any similar calculation online. I really appreciate if someone could help me out.
Hint: Split the integral into integrals over $(y, \infty)$ and $(-\infty,y)$ instead of $(0, \infty)$ and $(-\infty,0)$.