Improper Integrals: Solve ∫-∞ to ∞ e^-|x| dx

[Panphobia]

Homework Statement

\int_{-\infty}^{+\infty} e^{-\left|x\right|} \,dx

The Attempt at a Solution

So I know you are supposed to split this integral up into two different ones, from (b to 0) and (0 to a) where b is approaching - infinity, and a is approaching +infinity, but how would I take that antiderivative? Since the absolute value has a piecewise derivative.

 

[Saitama]

Homework Statement

\int_{-\infty}^{+\infty} e^{-\left|x\right|} \,dx

The Attempt at a Solution

So I know you are supposed to split this integral up into two different ones, from (b to 0) and (0 to a) where b is approaching - infinity, and a is approaching +infinity, but how would I take that antiderivative? Since the absolute value has a piecewise derivative.

Hi Panphobia!

Do you see you can write it as $\int_{-\infty}^0 e^x\,dx +\int_0^{\infty} e^{-x}\,dx$?

 

[Panphobia]

I totally missed that, but yea I see it now, thanks!