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It is well known that $$ \int \frac{f'(x)dx}{f(x)}= \int d \log f(x)=\log f(x) + C $$

In my work I came across the following case: $$ \int \frac{(f'(x))^2dx}{f(x)} $$

I wonder if any interesting approach exists in this situation

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Probably not. For example, if $f(x)=e^{x^2}$, then $(f')^2/f=4x^2e^{x^2}$, which has no elementary antiderivative. – Gerry Myerson Oct 25 '12 at 12:17
up vote 1 down vote accepted

Integrating by parts gives,

$$ \int \frac{f'(x)f'(x)dx}{f(x)} = f'(x)\ln(f(x)) - \int f''(x)\ln(f(x))dx \,.$$

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which makes things even more complicated, but thanks anyway. – Alex Oct 25 '12 at 20:37

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