Inverse DTFT Problem Having trouble finding the inverse DTFT of $\ X(\ e^{j \omega}) = \frac{3 - \frac{1}{4} e^{-j\omega}}{1 - \frac{1}{4} e^{-2j\omega}}  $

Given the IDFT of $Xe^{j \omega}$ as : 

$x(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\ e^{j \omega}) \cdot e^{j \omega n} d\omega$

Tried long division as an option and reduction to partial fractions but not to any good so far.

Not sure on how to proceed.
You can use partial fractions first to expand the $X(e^{j\omega})$ as

$$X(e^{j\omega})=\frac{\tfrac{7}{4}}{1+\frac{1}{2}e^{-j\omega}}+\frac{\tfrac{5}{4}}{1-\frac{1}{2}e^{-j\omega}} $$.

Then taking the inverse DTFT you get $\tfrac{7}{4}(-\frac{1}{2})^{n}u(n)+\tfrac{5}{4}(\frac{1}{2})^{n}u(n)$