Laplace transform of a signal over t

Question

I tried to indentify a formula that is appropriate for computing the Laplace transform of $$f(t)=\displaystyle\frac{\cos 2t-\cos 3t}{t}$$ but I couldn't find one. Give me a suggestion please. Thanks, Alex.

Answer 1

$$\begin{align*} \int\limits_s^\infty G(u)\cdot\text du &=\int\limits_s^\infty\int\limits_0^\infty g(t)e^{-ut}\cdot\text dt\cdot\text du \\&=\int\limits_0^\infty g(t)\int\limits_s^\infty e^{-ut}\cdot\text du\cdot\text dt \\&=\int\limits_0^\infty g(t)\left.\frac{e^{-ut}}{-t}\right|_s^\infty \cdot\text dt \\&=\int\limits_0^\infty \frac{g(t)}te^{-st}\cdot\text dt \\&=\mathcal L\left\{\frac {g(t) }t \right\} \end{align*}$$