Problem with NIntegrate in NonlinearModelFit

Question

We are receiving many error messages when using NIntegrate with NonlinearModelFit. Here is a much-simplified version of the code. It arrives at the correct answer after several messages saying that it is stopping the integration. In the real version, which is fitting Beta distributions, the situation is similar but more complicated, and it is not clearly getting the best fit.

data = {{0, 0}, {1, 1}, {2, 4}}

cnet[x_, b_, c_] := NIntegrate[b + 2 c u, {u, 0, x}]

Print[cnet[0, 0, 1]]
Print[cnet[1, 0, 1]]
Print[cnet[2, 0, 1]]
Print[cnet[3, 0, 1]]

Chop[NonlinearModelFit[data, cnet[x, b, c], {b, c}, x]]

Answer 1

I have seen this NIntegrate::nlim: u = x is not a valid limit of integration. >> message with my code generated from previous versions of Mathematica. The reason for this message comes from upgrades within the NIntegrate code to be more efficient and work more quickly. Previous versions allowed a symbol or a numeric value as input. If you use the ?NumberQ variable type explicitly, this should resolve your problem. It is kind of like type-casting your variables. The following code

Clear[data, cnet] 
data = {{0, 0}, {1, 1}, {2, 4}}

cnet[x_?NumberQ, b_?NumberQ, c_?NumberQ] := NIntegrate[b + 2 c u, {u, 0, x}]

cnet[0, 0, 1]
cnet[1, 0, 1]
cnet[2, 0, 1]
cnet[3, 0, 1]

Chop[NonlinearModelFit[data, cnet[x, b, c], {b, c}, x]]

Should give you output with no complaints.