What does it mean if I was told to "check if Leibniz's rule conditions are met in suitable rectangle"?
If that is necessary, I was asked about this function: $I(a)=\int_{0}^{1} \frac{dx}{1+ae^x} $
What are those conditions?
Note: I had a similar question which was written completely wrong (which made confusion) so I'm deleting that one and replacing in with this
As you have integral with parameter, then I suppose, that you are interested in derivative of $I(a)$. One classical result is, that if we consider integrand function $f(x,a)=\frac{1}{1+ae^x}$ on some rectangle, for example, $P=[0,1] \times [c,d] $ and $a \in [c,d]$. If $f$ and $\frac{\partial f}{\partial a}$ are continuous on $P$, then $I(a)$ is differetiable on $[c,d]$ and $$\frac{dI}{da}= \int_{0}^{1} \frac{\partial}{\partial a}\frac{ x}{1+ae^x}dx$$
So, now is essential where you want to consider $a$. Most dangerous/interesting is point $a=-1$.