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See continuous Fourier transform for more information, including a table of transforms, discussion of the transform properties, and the various conventions.A generalization of this transform is the fractional Fourier transform, by which the transform can be raised to any real "power".
Nevertheless, I thought it would be instructive to take a look at the signal $y(t)$, so I evaluated it numerically for the parameters $k=0.05$ and $\omega_0=\pi/10$.
The following figure shows the result: The green curve is the input signal $x(t)$ and the blue curve is the filtered signal $y(t)$.
It can be evaluated numerically using the exponential integral $\text(x)$, or, alternatively, the sine and cosine integrals $\text(x)$ and $\text(x)$.
So I don't think that the purpose of the exercise was to actually compute the convolution, but its purpose was probably to come up with a qualitative description of what is going on (exponential signal filtered by an ideal lowpass filter).
We'll save the detailed math analysis for the follow-up.
This isn't a force-march through the equations, it's the casual stroll I wish I had. Check out for a great tool to draw any shape using epicycles.
Rather than jumping into the symbols, let's experience the key idea firsthand.
Here's a plain-English metaphor: how any pattern can be built with cycles, with live simulations. moment and intuitively realize why the Fourier Transform is possible.
The Fourier integral of $x(t)$ can be easily computed: $$X(j\omega)=\int_^e^e^dt = \frac$$ The Fourier transform of $w(t)$ should be familiar because it is an ideal lowpass filter.
In the question there was some confusion concerning the definition of the Sinc function.