A computer gives the answer as $\displaystyle -\cos t$, but by hand, I seem to be getting a non-converging integral. What am I doing wrong?

$\displaystyle \sin t * u(t) = \int_{-\infty}^\infty \sin \tau \cdot u(t - \tau) d\tau$

Since $\displaystyle t > \tau$ for result to be non-zero, we have:

$\displaystyle = \int_{-\infty}^t \sin \tau d\tau$

$\displaystyle = (- \cos \tau)|_{-\infty}^t$

Which does not converge.