How can I solve this integral equation using Laplace transform?
$${\int\limits_0^{\infty}\ }\frac{e^{-t}(1-\cos t)}{t}\operatorname d\!t$$
Knowing that $$ \mathcal{L}\{\cos t\} = \frac{s}{s^2+1} $$
I think I can start by taking limits:
$$\lim_{b \rightarrow \infty} {\int\limits_0^{b}\ }\frac{e^{-t}(1-\cos t)}{t}\operatorname d\!t$$
ant then apply the shortcut of $$\mathcal{L}\{\cos t\}$$
but I don't know how to continue. Any help will be appreciated.