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.
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.