5
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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:

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

  • 0
    Yes. And $\int (f'(x))^2 dx$?2012-07-04
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    To write the solution without an integral is big and deep problem for mathematics.2012-07-04
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    @Mathlover, reference please?2012-07-04
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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.pdf2012-07-04
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    @Mathlover, that's a nice observation, thanks.2012-07-04

2 Answers 2

2

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

  • 0
    Yes. And $\int (f'(x))^2 dx$?2012-07-04
  • 2
    In general, nothing: it is what it is.2012-07-04
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    This isn't *Jeopardy!*2012-07-04
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    @draks What is math.stackexchange?2012-07-04
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Using by parts $\int f(x)f''(x)dx= f(x)f'(x)-\int f'^2(x)dx$.