Evaluate convolution integral

Question

Can someone tell me if I am calculating this integral correctly.

Answer 1

Don't you make a mistake at the 3th line?

You suddenly multiply $e^{At}B\delta(\tau)$ with the 1/A while it's a subtraction

Answer 2

if $\beta$ is constant you can just use the sifting property of the delta function which means you will sift $e^{-\tau}$ to 0 surely resulting in $\beta e^{At}\cdot [1]^{t}_{0}$ . You cant take $\delta(\tau)$ out of the integral since it is not constant.

Siftin property if T is within the range [a,b]:

\begin{equation} \int_{a}^{b} f(x) \delta(x - T) dx = [f(T)]_{a}^{b} \end{equation}