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