How to integrate absolute function

Question

I have this absolute e-function, but I don't know how to calculate the integration

$$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$

Any idea?

Answer 1

Observe that $e^{\frac{1}{2}j\omega |x|}$ is even so, by symmetry, we have:

$\displaystyle\int^2_{-2}e^{\frac{1}{2}j\omega |x|} dx = 2\displaystyle\int_0^2 e^{\frac{1}{2}j\omega x} dx$

Where we have dropped the moduli given that the integration is now taking place over an interval where $x$ is non-negative.

Answer 2

The integrand is symmetric under $x{\rightarrow}-x$ so we can restrict the integration interval to $[0,2)$ and putting a factor 2 in front. Now $|x|$ can be replaced by $x$ and the rest is straightforward.