Question

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=t^{2} $$

Short answer

The Laplace Transform of the function \(f(t)=t^{2}\) is \(F(s)=\frac{2}{s^{3}}\)

Step-by-step solution

Setting up the integral for Laplace transform

First, substitute \(f(t)\) with \(t^{2}\) in the Laplace transform formula \(F(s)=\int_{0}^{\infty} e^{-st}f(t) dt\). It results in: \[F(s)=\int_{0}^{\infty} e^{-st}t^{2} dt\]

Performing the integration by parts

Integration by parts is a suitable approach for this integral. The method requires choosing two parts of our integrand to be u and dv. Let us choose \(t^{2}\) as u and \(e^{-st}\) as dv. Compute du by differentiating u, and compute v by integrating dv. Hence, \(du=2t dt,\, dv=e^{-st}dt,\, v=-e^{-st}/s\). Then you can use the integration by parts formula: \[\int u\,dv = uv - \int v\,du\]. Part before minus is easy to solve. To find the integral of \(v\,du\) use the same method for multiplying polynomial and exponential again. The final step would be done by simple integration.

Checking the existence of improper integral

At last consider the limit for integral with \(t^{2}\) multiplied by exponential that tends to zero. Integrating from 0 to infinity, confirm the existence of the integral by the value converging to some real number. After that perform the limiting process with the boundary values \(0\) and \(\infty\) to come out with the final expression of the Laplace Transform.