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This was one of the later questions in my tutorial which I didnt reach in time. Answers for tutorials aren't posted online however so I tried working through this alone but quickly got stuck

$\displaystyle \hat{f}(w) = \frac{exp(\pmb{i}w)}{(2+3\pmb{i}w)} $

We're not really expected to calculate the integral but rather 'guess' the inverse transform by using linearity and well know properties of the fourier transform.

I know that $\displaystyle f(t) = e^{-at} -> \hat{f}(w) = \frac{1}{a+iw} $ so a = 2 in this case however I've got no idea what to use to get the 3iw down bottom or the exponential up top. Any help is appreciated.

Brodudedoodebrodude
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It's better to write the formula as $$\hat{f}(w) = \frac13 \frac{\exp(\pmb{i}w)}{((2/3)+\pmb{i}w)}$$ because now $e^{-at}u(t)$ (Heaviside function is also needed here) with $a=2/3$ gets you pretty close. The factor of $1/3$ can be easily attached: the transform is linear.

To get the factor of $\exp(\pmb{i}w)$, use the time-shifting property of the Fourier transform.