I'm trying to find the inverse laplace of the following function by using convolution.
$$\mathcal{L^{-1}}(\frac{s}{(s+1)^2})$$
What I did was to separate into:
$$\mathcal{L^{-1}}(\frac{s}{s+1}\,\frac{1}{s+1})=\mathcal{L^{-1}}[(1-\frac{1}{s+1})\,\frac{1}{s+1}]$$
Then:
$f(t)=\mathcal{L^{-1}}[1-\frac{1}{s+1}]=\delta(t)-e^{-t}\\g(t)=e^{-t}$
$f\ast g(t)=\int_0^t{(\delta(s)\,e^{-s})\,(e^{s-t})\,ds}$
The second part of the integral is easy but I don't know how to deal with the Dirac integral. Is the first time I'm trying to solve an inv lapl using convolution theorem, I believe I could choose another one to start but I want to know how to deal with this one. I could use the fact that the derivative of $\mu$ is the Dirac function but then I don't know what's the constant $k$ in $\mu_k(s)$. Please I'd like some help, I don't know if I'm making a mistake since the beginning too.
I think you forgot a minus sign in your integral.
$$\left( 1-\frac{1}{s+1} \right)\frac{1}{s+1} \rightarrow \left(\delta(t)-e^{-t}\right)\ast e^{-t}=\delta(t) \ast e^{-t}-e^{-t}\ast e^{-t}$$
$$=e^{-t}-\int^t _0 e^{-\tau}e^{-t+\tau}d\tau=e^{-t}-e^{-t}\int^t _0 e^{-\tau}e^{+\tau}d\tau=e^{-t}-t e^{-t}$$