Signals and Systems - Taking Integral of Unit Step Function
I suspect \$ x_1(t) = e^{-2t} u(t)\$

Since \$u(t) = 0 \$ for \$t&lt;0\$, the next step simple changes the boundaries of the integration to reflect that the product shown is zero for all \$t&lt;0\$.  Once you do that, you can just drop the \$u(t)\$, since multiplying a signal  by one just returns the original signal.