Computation of $\int_a^b x f(x)\,\mathrm{d}x$, where $f$ is the log-normal density function

I'm trying to evaluate the following integral: $$\int\limits_a^b x f(x)\,\mathrm{d}x$$ where $f$ is the probability density function of the log-normal distribution.

If $a=0$ and $b=\infty$, the result is (I think) the mean, $\exp({\mu +\sigma ^{2}/2})$. Looking at the PDF, the result should be some simple function involving $\Phi$ or $\phi$, the CDF or PDF of the normal distribution; but I can't get it to work. I'm sure this is a standard result, but ... can anyone point me in the right direction?

• Ok, further searching pointed me to the answer embedded in this similar question here... link – user219142 Apr 25 '17 at 14:30
• You can find a cute trick to get the expected value here: math.stackexchange.com/questions/1065914/… – mlc Apr 25 '17 at 14:44