I tried to solve the following to problems from Gaskil's book Linear Systems, Fourier Transforms, and Optics. But I'm struggling to get the right results. My experience with calculating convolutions is very limited. I wanted to see how this could be done using direct integration, the filter theorem makes the task trivial.
(a.)
$GAUS\left[\frac{x}{3}\right]\ast GAUS\left[\frac{x}{4}\right] = \int_{-\infty}^{+\infty}e^{-\pi\left(\frac{\alpha}{3}\right)^2}e^{-\pi\left(\frac{x-\alpha}{4}\right)^2}\mathrm{d}\alpha$
(b.)
$SINC\left[3x\right]\ast SINC\left[2x\right] = \int_{-\infty}^{+\infty}\left(\frac{\sin[3\pi\alpha]}{3\pi\alpha}\right)\left(\frac{\sin[2\pi(x-\alpha)]}{2\pi(x-\alpha)}\right)\mathrm{d}\alpha$