Monday Math 112

The Laplace Transform
Part 13: Transform of the Logarithm

Now, let us find the Laplace transform of the natural logarithm. From the definition of the Laplace transform,
.
Making the u-substitution u=st, we get
.

Now, one may recall that we found the remaining integral here:
, where γ is the Euler-Mascheroni constant. Thus,
.