Notation Concerning Improper Integral and the Absolute Value of an Integrand I have the following integral
$$\int_{- \infty}^\infty e^{-|x|} dx$$
Since the preimages $x$ determine the the images $e^{-|x|}$ for nonnegative and negatives preimages, I believe the integral must be evaluated in the following manner
$$\lim \limits_{c \rightarrow 0} \enspace \int_{- \infty}^{c} e^{-|x|} \enspace dx \quad + \quad \int_{0}^{\infty} e^{- |x|} \enspace dx$$
where $c$ obviously approaches $0$ without necessarily equalling $0$, so that $e^{- |x|} = e{-(-x)}$. So,
$$\lim \limits_{c \rightarrow 0} \enspace \int_{- \infty}^{c} e^{-|x|} \enspace dx \quad + \quad \int_{0}^{\infty} e^{- |x|} \enspace dx $$
$$= \lim \limits_{c \rightarrow 0} \enspace \int_{- \infty}^{c} e^{x} \enspace dx \quad + \quad \int_{0}^{\infty} e^{-x} \enspace dx $$
$$= 2$$
Also, I think it is wrong to write 
$$\int_{- \infty}^{\infty} e^{-|x|} \enspace dx \quad + \quad \int_{0}^{\infty} e^{- |x|} \enspace dx$$
Since the first integral includes the value $0$, causing discrepancies with $e^{-|x|}$.
This is more of a notation issue and I would like some insight. Thank you.

Notice, the given function $$f(x)=e^{-|x|}\implies f(-x)=e^{-|-x|}=e^{-|x|}=f(x)$$ hence $f(x)=e^{-|x|}$ is an even function
Now, using property of definite integral $\color{blue}{\int_{-a}^{a}f(x)dx=2\int_{0}^{\infty}f(x)dx}$, we get
$$\int_{-\infty}^{\infty}e^{-|x|}dx=2\int_{0}^{\infty}e^{-|x|}dx$$
Since, $|x|=x\ \forall \ \ 0\leq x$ hence 
$$2\int_{0}^{\infty}e^{-|x|}dx=2\int_{0}^{\infty}e^{-x}dx=2[-e^{-x}]_{0}^{\infty}$$ $$=2[-e^{-\infty}+e^{0}]=2[0+1]=2[1]=\color{red}{2}$$
