"inverse" of an integral

Question

I am reading an economics paper Melitz and Ottaviano (2008), but I find it hard to understand some math in the model.

The preference is given by :

, which implies a demand function:

Then the paper says the "inverse demand function can then be inverted to yield the linear market demand system for these varieties":

But what operations do they take in getting the function $q_i$?

I am very confused on dealing with the $Q_c$ variable, which is defined to be :

Thanks.

Answer 1

Actually, just take an integral on both sides of equation 2 w.r.t. $i$, and we get an expression of $Q_c$. Then plug the $Q_c$ back into equation 2.