I wouldn't call it integration, but knowing that the thermal conductivity is a constant, as well as the boundary temperature, you can calculate
And then there is the difference between heat flow and heat flux, the latter being the heat flow per unit area, hence the factor $A$.
Formally, you would not know that the temperature profile is linear, but you can derive it from Fourier's law. Basically, you assume that the heat flux on point 1 and point 2 are equal, which givees you the steady state form of the heat equation
which is integrated twice to obtain the linear profile.