Chapter 15.3, Problem 20E

To determine

The Fourier integral representation of $f\left(x\right)=e^{-\left|x\right|}$ using complex form $f\left(x\right)=\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }C\left(\alpha \right)e^{-i\alpha x}}d\alpha$ and show that the obtained solution is $f\left(x\right)=\frac{2}{\pi }{\displaystyle {\int }_0^{\infty }\frac{\cos \alpha x}{1+{\alpha }^2}}d\alpha$.