Is there some procedure for figuring this out or am I venturing into unsolved territory here?
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I don't think that $\int f'(x)^2 dx$ can be stated in terms of $f(x)$. – user1892304 May 19 '16 at 20:15
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It seems unlikely. Compare: $f(x)=\sqrt{x}$, $f(x)=x$, and $f(x)=x^2$. – almagest May 19 '16 at 20:15
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If $f(x)=e^{x^2}$, then $(f'(x))^2$ doesn't have an antiderivative in closed form in terms of the familiar functions of intro Calculus. – Gerry Myerson May 19 '16 at 22:51