Consider a relation defined on the set of polynomials, such that two polynomials are related if and only if their indefinite integrals are equal. Is this an equivalence relation?
I'm slightly confused with one thing in this question - since every function has infinitely many indefinite integrals, is something going wrong here? In case the answer is no, then I feel its both reflexive and symmetric, but I'm not sure how to prove its transitivity (or lack of it). So please help me with this question, so that I can strengthen my concepts further on relations.
As noted, the relation is painfully vague and ambiguous. After all,
$$\frac 1 2 x^2 + 2 \qquad \frac 1 2 x^2 + 3$$
are both indefinite integrals for $f(x) = x$. Thus within a certain frame of mind, a function doesn't even share an antiderivative with itself. So it would be better to define the notion of this relation as so:
$$f \sim g \iff \int f(x)dx - \int g(x)dx = \text{constant}$$
That is possibly more in line with the intention of the exercise.
...granted, this question is quite old, so I imagine you don't need help now. But hopefully this helps someone in the future, and, if nothing else, gets this question out of the unanswered queue. Posting as Community Wiki since I don't have much to add.