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B Rule to integrate a function with respect to its derivative

  1. Sep 10, 2017 #1
    Hello all, I was just wondering if there is any rules for integrating a function with respect to it's own derivative.

    That is to say ##\int _{ }^{ }f\left(x\right)d\left(f'\left(x\right)\right)## or ##\int _{ }^{ }yd\left(\frac{dy}{dx}\right)##

    Thank you in advance for your time :)
     
  2. jcsd
  3. Sep 10, 2017 #2

    fresh_42

    Staff: Mentor

    I came until ##\int f(x) d(f'(x)) = f(x)f'(x) - \int (f'(x)^2)dx## so its the integral of a function squared, which has no general solution without knowing ##f'(x)##.
     
  4. Sep 10, 2017 #3

    andrewkirk

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    Science Advisor
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    We can use substitution. Set ##u=f'(x)##. Then ##\frac{d(f'(x))}{dx}=\frac{du}{dx}=f''(x)## so in the integral we can replace ##d(f'(x))##, which is ##du##, by ##f''(x)dx##. That gives us:
    $$\int f(x)f''(x)dx$$
    Whether a closed form can be found for the integral depends on ##f##.
     
  5. Sep 11, 2017 #4
    Thank you both for your help :)

    I knew that ##\int _{ }^{ }f\left(x\right)d\left(f\left(x\right)\right)## has the general solution of ##\frac{1}{2}\left(f\left(x\right)\right)^2## regardless of what the function actually is. I was curious as to whether there would be a way to apply this while integrating with respect to the functions derivative instead, but it looks like there's no general solution for this.
     
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