Convolution integral with fraction expansion

Question

I have to solve this convolution integral $$ \int_{-\infty}^{+\infty} \frac{1}{\frac{1}{T_1} + i2\pi \tau } + \frac{1}{\frac{1}{T_2} + i2\pi (f- \tau) } d \tau $$ but I have a lot of problems with partial fraction expansions (my book tells to use this method ). First of all I can write it like this $$ \int_{-\infty}^{+\infty} \frac{A}{\frac{1}{T_1} + i2\pi \tau } + \frac{B}{\frac{1}{T_2} + i2\pi (f- \tau) } d\tau $$ and after I found least common denominator $$ A{\frac{1}{T_2} + i2\pi (f- \tau) } + B{\frac{1}{T_1} + i2\pi \tau }=1 $$ . I wrote 1 because in the first numerator i hadn’t any other components except for 1. Now I solved and I obtained $$ \tau ( - Ai2 \pi + Bi2 \pi ) + Ai2 \pi f + \frac{A}{T_2} + \frac{B}{T_1} - 1 = 0 $$ at this point I wrote that all the tau component is 0 , because the original numerator was 1. I don’t know if there’s something right however now I’m completely blocked :/