Finding Impulse Response for System?
\$h(t) = \int_{-\infty}^t \! \delta(T-2)e^{-(t-T)} \, \mathrm{d}T = e^{-(t-2)}u(t-2)\$

The delta "function" is zero except where the argument is zero, i.e., when T=2, where it has an area of 1.

So, if \$ t &lt; 2\$, the integral is zero.

If \$ t \ge 2\$, the integral equals the area of the delta function multiplied by the value of the exponential when T = 2.