What is $\exp(-3|x|)$ mean?

Question

In $\exp(-3|x|)$

While splitting the integral $x$ from $-\infty$ to $0$ and $0$ to $\infty$, the first integral is replaced by $\exp(3x)$ and the second integral by $\exp(-3x)$?

I would like to know how do we obtain the first integral as $\exp(3x)$? The point I am confusing is mod means absolute(or positive value)? and then how comes the negative $x$ values remains negative?

Answer 1

$$ \exp(-3|x|)=\exp(-3x),x\geq0,\exp(3x),x<0 $$

Answer 2

A nice way to understand $f(x)=e^{-|x|}$ is that you always need negative powers of exponential function. So for $x<0$ you take $f(x)=e^x$ and for $x>0$ you take $f(x)=e^{-x}$.