Can we find two functions $f$ and $g$ that are reasonably defined nontrivial(not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied?

$$ f( \left(\int_{0}^{t} g(x) \ \text{d}x\right))  = g( \left(\int_{0}^{t} f(x) \ \text{d}x\right)) $$

P.S.: I migrated this question from [here][1] on   Math.SE. I am sure this site hosts very knowledgeable mathematicians that keeping on migrating to another site is foolish. I felt a really good feeling for some time as nobody answered my question. But is usually the case that: "There is a general principle that a stupid man can ask such questions to which one hundred wise men would not be able to answer. In accordance with this principle I shall formulate some problems." Vladimir Arnold


  [1]: http://math.stackexchange.com/questions/18005/are-there-functions-satisfying-the-following-integral-condition