Evaluating $\int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}\,\mathrm{d}x $ I'm trying to figure out this integral but cannot figure out the right substitution 
$$
\int^\infty  _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}dx 
$$

Let
$$
\begin{align}
I(\alpha,k) 
&= \int^\infty _{-\infty} \frac{e^{-i  \alpha x}}{x^2 + k^2}\,\mathrm{d}x \tag{1} \\
&= \int^{\infty} _{-\infty} \frac{e^{i  \alpha x}}{x^2 + k^2}\,\mathrm{d}x \tag{2} \\
&= \frac12\int^{\infty} _{-\infty} \frac{e^{i \alpha  x} + e^{-i \alpha x}}{x^2 + k^2}\,\mathrm{d}x \tag{3} \\
&= \int^{\infty} _{-\infty} \frac{\cos \alpha x}{x^2 + k^2}\,\mathrm{d}x  \tag{4} \\
&= 2\int^{\infty} _{0} \frac{\cos \alpha x}{x^2 + k^2}\,\mathrm{d}x  \tag{5} \\
I(\alpha,k) 
&= \frac\pi {|k|} e^{-|\alpha|| k|}  \tag{6} 
\end{align}$$
Putting $\alpha = \dfrac ph$ and $k = a$ in $(6)$

$$I\left(\frac ph,a\right)  = \int^\infty _{-\infty} \frac{e^{-i  p x /h}}{x^2 + a^2}\,\mathrm{d}x
= \frac \pi {|a|} e^{-|a||p/h|}$$


$\text{Explanation (2) :}$ Set $x \rightarrow -x$
$\text{Explanation (6) :}$ Using 
$$\int_{0}^{\infty} \frac{\cos \alpha x}{x^2 + k^2}  dx = \frac{\pi e^{-k\alpha}}{2k}$$
