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Integrating derivatives of various orders

  1. May 25, 2013 #1
    I'm trying to figure out the general solution to the integral ##\int \frac{d^ny}{dx^n} \, dy##, where n is a positive integer (Meaning no fractional calculus. Keeping things simple.).

    So far, I have been working with individual cases to see if I can establish a general pattern and then try a proof by induction.

    So far, I have

    $$\int\frac{dy}{dx} \, dy = \frac{d}{dx}\left[y\frac{dy}{dx}\right] + C = y\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2 + C \\ \int\frac{d^2y}{dx^2} \, dy = \frac{1}{2}\left(\frac{dy}{dx}\right)^2 + C$$

    and I'm working on n=3.

    However, for n=3, I get

    $$\int \frac{d^3y}{dx^3} \, dy = \int \frac{d^3y}{dx^3}\frac{dy}{dx} \, dx = \int \frac{d}{dx}\left[\frac{d}{dx}\right] u \, dx$$

    and then I don't know what to do. Any suggestions?

    I've noted that ##\displaystyle \int\frac{d^3y}{dx^3} \, dy = \frac{d}{dx}\left[y\frac{d^3y}{dx^3}\right] = \frac{dy}{dx}\frac{d^3y}{dx^3}+y\frac{d^4y}{dx^4}##?
    Last edited: May 25, 2013
  2. jcsd
  3. May 26, 2013 #2
    Do you actually get back the integrand if you differentiate your right hand side with respect to y?

    (Besides that, I think your problem is easiest to answer using the Leibniz integral rule.)
  4. May 26, 2013 #3

    Sorry to say : There are some mistakes.
    Did you try to test your équations with simple functions, for example y(x)=ax+b ?
    Last edited: May 26, 2013
  5. May 26, 2013 #4
    Nevermind. Fixed the problem. Really bad math day.

    Feel free to close the thread.
    Last edited: May 26, 2013
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