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I have a curiosity. If

$\int f(x) f'(x)dx=\int f(x) df(x)=\frac{\left(f(x)\right)^{2}}{2}+C$

what is the result of:

$\int f(x) f''(x)dx$

Mark
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  • Yes. And $\int (f'(x))^2 dx$? – Mark Jul 04 '12 at 10:59
  • To write the solution without an integral is big and deep problem for mathematics. – Mathlover Jul 04 '12 at 11:00
  • @Mathlover, reference please? – lhf Jul 04 '12 at 11:05
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    @lhf:I do not know any technics to solve $\int f'(x)^2 dx $ without endless series. if you have a closed form solution of $\int f'(x)^2 dx$ and also you would have a closed form of $\int 4x^2e^{2x^2} dx$ when $f(x)=e^{x^2}$ with elemantary functions. They are related to each other. Please see http://math.stanford.edu/~conrad/papers/elemint.pdf – Mathlover Jul 04 '12 at 11:40
  • @Mathlover, that's a nice observation, thanks. – lhf Jul 04 '12 at 11:57

2 Answers2

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Using by parts $\int f(x)f''(x)dx= f(x)f'(x)-\int f'^2(x)dx$.

Aang
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$$ f(x)f'(x)-\int f'(x)^2 \, dx \ ? $$

Siminore
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