Mathematica Mistake? Solve Puzzling Integration Issue

[keniwas]

There is something pretty strange going on with mathematica right now.

When I do the following function

Integrate[(n x + 1)^2,x]

I get the result
$$\frac{(1+nx)^3}{3n}$$
Expanded this is
$$\frac{1}{3 n}+x+n x^2+\frac{n^2 x^3}{3}$$

However this is not the result I get if I integrate the individual parts of the expansion
$$(nx+1)^2=1+2nx+n^2x^2$$
and add the results of the integrals
i.e.

Integrate[n^2 x^2, x] + Integrate[2*n*x, x] + Integrate[1, x]

which gives me
$$x+n x^2+\frac{n^2 x^3}{3}$$

Please tell me I am missing somthing obvious and mathematica isn't making a mistake?

 

[CompuChip]

If you integrate w.r.t x, then 1/(3n) is a constant. You have not given integration boundaries, so this is allowed. In other words, since D[1/(3n), x] = 0, both results differentiate back to $(nx + 1)^2$.

 

[keniwas]

Granted it is allowed, why would it choose such an esoteric constant? Like you said I didn't provide it any boundary conditions, so what's so special about this constant? It seems rather unusal to me, and its not a behavior I have seen from the software before when performing integration that I am aware of.

 

[Dale]

The constant is not particularly esoteric. It is a direct result of applying the chain rule to (n x + 1)^2. In any case, as mentioned above integration is only defined up to a constant, so there is no mistake here other than the fact that the constant is not explicitly mentioned (which I think it should do).