Laplace transform and Integral $f(t)$

Question

I calculate Laplace transform. $$f\left( t\right) =\dfrac{e^{λ_1t}-e^{λ_{2}t}}{t}$$ $$λ_1,λ_2>0$$

---

I think

$$\int _{0}^{\infty }f\left( t\right) e^{-st}dt=\int _{0}^{\infty }\dfrac{e^{\lambda _{1}t}-e^{\lambda _{2}t}}{t}e^{-st}dt$$ $$=\int _{0}^{\infty }\dfrac{e^{\left( λ_{1}-s\right) t}}{t}dt-\int _{0}^{\infty }\dfrac{e^{\left( λ_{2}-s\right) t}}{t}dt$$

I don't know what to do next. Please tell me hint.

Answer 1

There is a property of the Laplace transform where $\mathscr{L}\Big\{\frac{f(t)}{t}\Big\}= \int_{s}^{\infty}F(u)\,du$ where $F(s)=\mathscr{L}\{f(t)\}$