70 Examples of Laplace transforms

$$\text{Example}\mathit{\hspace{1em}\hspace{1em}}ℒ(x^2;s)=\frac{2}{s^3}$$

To see this, consider $R>0$ and integrate by parts

$${\int }_0^R{x}^2{e}^{-sx}𝑑x={\left[\frac{x^2{e}^{-sx}}{-s}\right]}_0^R+\frac{2}{s}{\int }_0^{R}x{e}^{-sx}𝑑x$$

$$={\left[\frac{x^2{e}^{-sx}}{-s}\right]}_0^R+{\left[\frac{2x{e}^{-sx}}{-s^2}\right]}_0^R+\frac{2}{s^2}{\int }_0^R{e}^{-sx}𝑑x$$

$$={\left[\frac{x^2{e}^{-sx}}{-s}\right]}_0^R+{\left[\frac{2x{e}^{-sx}}{-s^2}\right]}_0^R+{\left[\frac{2e^{-sx}}{-s^3}\right]}_0^R$$

$$=-\frac{R^2{e}^{-Rs}}{s}-\frac{2R{e}^{-Rs}}{s^2}-\frac{2e^{-Rs}}{s^3}+\frac{2}{s^3}$$

$$\to \frac{2}{s^3}\mathit{\hspace{1em}\hspace{1em}}\text{as}\mathit{\hspace{1em}\hspace{1em}}R\to \mathrm{\infty }.$$