How to integrate an integral which contain an oscillatory term $e^{\imath x\cdot p}$

Question

I need to evaluate a integral $$f(p)=\int_0^\infty dx \frac{x^2}{2\sqrt{x^2+a^2}}e^{-\imath x\cdot p}, $$I tried a lot, but unable to find a method to integrate due to the presence of oscillatory term $e^{\imath x\cdot p}$.

Please give some ideas to evaluate this integral.

Answer 1

This improper integral does not converge as asymptotically the absolute value of the integrand namely $|f(x)| \sim x, x\in(0,\infty).$ Hence, $\int_{0}^{\infty} f(x) dx$ diverges.

Answer 2

I agree with Z Ahmed. Here is the case $a=0$, where it can be done in closed form.

For $p=1$ the graph of $\text{Re }\int \frac{x}{2} e^{-ix}\;dx$ looks like

and the limit does not exist at $+\infty$.
For $p=3$ the graph of $\text{Re }\int \frac{x}{2} e^{-i3x}\;dx$ looks like

and the limit does not exist at $+\infty$. You can guess what the graph looks like for other real values of $p$.
Now it does converge for $\text{Im}\;p < 0$. Here is $p=-1+i\, 2.5$