Question

f(t)=t. find directly the laplace transforms of the function

Answer 1

1/s^2

Answer 2

because the laplace transform of t^n=n!/s^(n+1)

Answer 3

this is from laplace transform tables..if you want to do the integral its the same result...its just simply a looot more of algebra

Answer 4

$$\int_0^\infty te^{-st}\,\mathrm{d}t$$Use parts on \(\int te^{-st}\):$$\int te^{-st}\,\mathrm{d}t=-\frac{t}se^{-st}-\int e^{-st}\mathrm{d}t=-\frac{t}se^{-st}-\frac1{s^2} e^{-st}$$
Our improper integral then reduces to:$$\lim_{a\to\infty}\left[-\frac{t}se^{-st}-\frac1{s^2}e^{-st}\right]_{t=0}^{t=a}=\lim_{a\to\infty}\left(-\frac{a}se^{-as}-\frac1{s^2}e^{-as}+\frac0se^0+\frac1{s^2}e^0\right)=\frac1{s^2}$$