How Do You Integrate -e^(-|x|) Over All Real Numbers?

[Cocoabean]

1. Homework Statement and equations
Find $$\int$$ -1e^-|x| between negative infinity and positive infinity.

2. The attempt at a solution

So I tried to just integrate using laws for logarithmic equations and got:
e^-|x| between negative infinity and positive infinity.
Of course this leaves me with:
e^-∞-e^-∞ = 0
I know the answer is -2, but I am not sure how to get there. Help would be much appreciated.

 

[danago]

Try splitting the integral into two separate integrals over two separate domains.

Use the fact that |x| = -x when x<0 and |x|=x for x>0.

 

[Cocoabean]

Thanks danago, I think i figured it out. I split the integral and added the integral of the function from 0 to infinity with -infinity to 0. The first one came out to -1 and the next came out to 1 respectively, giving me -2. I fail to see how I should have caught this before though since the function is defined at 0. I guess you could see that is symmetric and just double the integral from 0 to infinity?