Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$

Question

Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$ using residues.

So I have a theory how to calculate $PV \int_{-\infty}^{\infty} f(x)e^{iax}dx$ a>0, but I don’t know how to transform my integral to this form.

Answer 1

$$\begin{align}I&=\int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx\\&=-\frac{1-e^{iax}}{x}\bigg|_{-\infty}^\infty-ia\int_{-\infty}^{\infty} \frac {e^{iax}}{ x}dx\\&= -ia\int_{-\infty}^{\infty} \frac {e^{iax}}{x}dx\\&= -ia\times\pi i\operatorname{Res}_{z=0} \frac{e^{iaz}}{z}\\& = -ia\times\pi i e^{iaz} \bigg|_{z=0}\\& = \pi a\end{align}$$