Solve Laplace Transform: t^m = m!/s^m+1

[catcherintherye]

I am trying to show that L[t^m] = m!/s^m+1, unfortunately I can not understand why integral from 0 to inf of (t^m)(e^-st)dt = (-d/ds)^m. integral 0 to inf of e^-stdt... ?

 

[marcusl]

You can exchange the order, that is, take -d/ds inside the integral, since they are both linear operations. It takes down a factor t from the exponential. Do it m times and get t^m.

 

[HallsofIvy]

If m is a positive integer, use induction together with integration by parts:
$$\int_0^\infty t^m e^{-st}dt$$
Let u= t^m, dv= e^-stdt. Then du= m t^m-1 and v= -1/s e^-st. The integral becomes
$$-\frac{m}{s}\int_0^\infty t^{m-1}e^{-st}dt$$
since uv= 0 at both ends.