Tsin(aT+b) cannot be integrated from T=minus infinity, because the integral function contains summed term Tcos(aT+b). It doesn't get eliminated. It's undefined (as a limit value) at T=minus infinite. So, the formula of x(t) doesn't define a function. The starting point is impossible and the rest is meaningless.
It's possible that by applying different Fourier transform rules one gets a formula which contains delta function or its higher powers. I haven't tried it. It seems to need something like a derivative of the delta function which is two gears beyond classical analysis. The delta function alone becomes from the point frequency. Its derivative becomes from multiplying the sine by T in Tsin(aT+B). It's presented in Fourier transform collections like this:
The formula is taken from this website http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
Another formula for the symbol game can be found from the same collection. It's this: