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.).(adsbygoogle = window.adsbygoogle || []).push({});

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?

Edit:

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}##?

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# Integrating derivatives of various orders

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