Is is possible to obtain a closed form solution for the following integral?
$$\int_{t_{1}}^{t_{2}} \frac{1}{1 + Ae^{-kt}}\mathrm dt$$
You may recognize that the integrand is a variation of the logistic equation.
My question is motivated from an article I recently read in the 1996 MAA publication, VITA MATHEMATICA entitled ``How Many People Ever Lived,'' in which the author develops an argument by integrating the exponential function $N=ae^{bt}$. I would like to perform a few calculations using the more realistic logistic model; but first, I need to be able to integrate it.
Thus, the question: Is it possible to do this directly; if so, how?
Or, must one resort to numerical methods?
$$I = \int_{t_1}^{t_2}\dfrac{1}{1+Ae^{-kt}}dt=\int_{t_1}^{t_2}\dfrac{e^{kt}}{e^{kt}+A}dt = \frac1k\int_{t_1}^{t_2}\dfrac{1}{e^{kt}+A}d(e^{kt}) = \frac1k\left[\ln\bigg|\frac{e^{kt_2}+A}{e^{kt_1}+A}\bigg|\right]$$