I am a taking signals and systems course and my professor posted the solutions to our latest homework and I am trying to understand one of the steps. Below is the solution. Bracketed in red is what I am trying to understand.
Why does u(t) disappear when you move the bounds of the integral to infinity to zero? I am just trying conceptually understand why instead of just accepting it as fact.
The unit step function changes from 0 to 1 at x=0. The integral of the unit step from -infinity to 0 is 0. Therefore you move the lower limit up to 0 and remove the unit step function.
I suspect \$ x_1(t) = e^{-2t} u(t)\$
Since \$u(t) = 0 \$ for \$t<0\$, the next step simple changes the boundaries of the integration to reflect that the product shown is zero for all \$t<0\$. Once you do that, you can just drop the \$u(t)\$, since multiplying a signal by one just returns the original signal.
