Convolution Integral with Unit Step Function

Question

I'm struggling to find the integral of given function:

Given $\ x(t) = u(t+1) + u(t-1) + u(t+2)$ and $\ h(t) = e^{-t} cos(t) u(t) $, I need to find $\ y(t) = dx(t)/dt * h(t) $

The "*" sign is convolution.

I found that $\ dx(t)/d(t)$ is $\ δ(t+1) + δ(t-1) + δ(t+2)$

Please help me to solve this question. I couldn't tell the limits of my integral and how to solve it with $\ δ(t)*h(t)$ type of integral.

Answer 1

Look at the sampling (also known as sifting) property of Dirac delta.

That is

$$\boxed{f(t)*\delta(t-T)=\cdots =f(t-T)}$$

Hence, your answer is simply $$y(t)=h(t+1) + h(t-1) + h(t+2)$$