Let $t$, $l_1$and $l_2$ be some positive constants. I am trying to calculate the following integral: $$f(a)=\int_0^t \large e^{-l_2\frac{(a+1)(t-x)}{(a-(t-x))}} e^{l_1x}\small \ dx$$ where $a$ is positive. It can be noticed that, if I am not mistaken, $a$ should be $\ge t$ because otherwise the integral diverges; I made this observation using some numerical evaluations of the integral.
My main goals are to:
(i) study how $f(a)$ changes with $a$, and (ii) find $f(a)$ (or a lower bound on $f(a)$)
(i) I noticed that $g(a)=\exp[-l_2\frac{(a+1)(t-x)}{(a-(t-x))}]$ increases with $a$, and I think (using some plots) that it has an inflexion point.
Since $g(a)$ increases and $f(a)$ is nothing but a surface computation, we can claim that $f(a)$ also increases (?). However, I don't think that
I can claim that $f(a)$ has also one inflexion point; am I correct ?
(If so, this leads us to the second question).
(ii) Is it possible to find $f(a)$? if not, is there any (tight) lower bound on $f(a)$?