Consider the integral $\int f(x)\, dx$ where $f(x)=e^{-x^2}$, it has been proven that there is no antiderivative made from elementary functions, but is there a way to prove that there is not such integral $$\int (g(x(t),y(t))) · r'(t) dt = \int f(x) dx$$ for some appropriate function $g(x,y)$ and some appropriate curve $r(t)=(x(t),y(t))$? If it is possible, then maybe $r'(t)$ could help with a $u$-sub, to find an antiderivative?
2026-03-28 20:57:03.1774731423
Is it possible to prove that a function doesn't have an antiderivative?
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