Aren't physics integrals beautiful?

We'll use the fact that $\int _{ -\infty }^{ \infty }{ \sin { \left( x \right) } } dx\quad =\quad 0$ and also ${ e }^{ ix }=\cos { \left( x \right) +i\sin { \left( x \right) } }$

First we see that the integrand composes of a imaginary part and a real part. However the imaginary part can be thought of as the sum of sine and cosine as shown earlier. Also, from the fact that the sin(x) is a odd function, the integral of the sin(x) term will go to zero thus, we can say that the integral becomes

$Re\left[ \int _{ -\infty }^{ \infty }{ exp\left( \frac { i{ p }(x-{ x }_{ 0 }) }{ \hbar } \right) } exp\left( -\frac { \left| p \right| }{ { p }_{ 0 } } \right) dp \right]$

Also the rest of the terms are even, thus leaving us with

$2Re\left[ \int _{ 0 }^{ \infty }{ exp\left( \frac { i{ p }(x-{ x }_{ 0 }) }{ \hbar } \right) } exp\left( \frac { p }{ { p }_{ 0 } } \right) dp \right]$

Next, is the integration, particular tricks are used here, just standard stuff

$2Re\left[ \frac { 1 }{ \left( \frac { i(x-{ x }_{ 0 }) }{ \hbar } \right) } +\frac { 1 }{ { p }_{ 0 } } \right] \$

Now that's disgusting, lets make it neater

$2Re\left[ \frac { { p }_{ 0 }\hbar }{ i{ p }_{ 0 }(x-{ x }_{ 0 })+\hbar } \right]$

We're pretty much done, just take the conjugate, are there you go.