Entropy, denoted as H, is:
$$ H = -\int_a^b p\ln(p) dx $$
where the range a to b is some arbitrary boundary and where p is given by the classic:
$$ p(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left( \frac{x-\mu}{\sigma}\right)^2} $$
Here is what I have tried so far to analytically solve this integral:
1) Integration by parts (too messy to write all out here), but quickly realized I needed a fancy substitution, thus leading me to:
2) Convert from x-space (where the range is a to b) to t-space (where the range is $\alpha$=$\frac{a-\mu}{\theta}$ to $\beta$=$\frac{b-\mu}{\theta}$) and, in general, $t$=$\frac{x-\mu}{\theta}$.
I think I'm on the right track (the $\theta$ in the denominator nicely cancels using the substitution approach) but I got stuck going from x-space to t-space and back again (i.e., I'm in xt-space purgatory and I want out!). Can anyone help me make more headway into #2 or perhaps suggest an alternative approach?