# A laplace transform

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
Gold Member
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
$$\int_0^\infty t^m e^{-st}dt$$
$$-\frac{m}{s}\int_0^\infty t^{m-1}e^{-st}dt$$