Integral of absolute value: $\int_{-\infty}^\infty {e^{-\frac{2}{b}|x - \mu |}}dx$

Question

I am stuck trying to integrate

$$\int_{-\infty}^\infty {e^{-\frac{2}{b}|x - \mu |}}dx$$

Incidentally, I'm interested in solving equation (5) in this paper using the Laplace distribution. I just got stuck integrating an absolute value.

Answer 1

Hint. We have $$\int_{-\infty}^\infty e^{-\frac2b|x-\mu|}\,dx =\int_{-\infty}^\mu e^{\frac2b(x-\mu)}\,dx +\int_{\mu}^\infty e^{-\frac2b(x-\mu)}\,dx\ ,$$ and I think you should find both of those integrals pretty easy.

Answer 2

You may just perform the change of variable $u=x-\mu$, giving

$$ \int_{-\infty}^\infty e^{-\frac2b|x-\mu|}\,dx =\int_{-\infty}^\infty e^{-\frac2b|u|}\,du=2\int_0^\infty e^{-\frac2bu}\,du=b. $$