Impulse Response of a circuit
Note that sin(t)/t is a function called a "sinc".  Convolving with a sinc yields some very special properties.  These are best to consider in the frequency domain.  Look up something called a "dual".

You should go thru the math once, but a sinc and its dual is something you should remember outright.  Again, go look this up.  These are worth knowing without having to do the calculations each time.

In this case, convolution by a sinc yields a special frequency characteristic.  However, due to how duals work, this other function in the time domain results in a sinc in frequency domain.  Again, this is one pair really worth knowing.