Integrating derivatives of various orders

[Mandelbroth]

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?

Edit:
I've noted that $\displaystyle \int\frac{d^3y}{dx^3} \, dy = \frac{d}{dx}\left[y\frac{d^3y}{dx^3}\right] = \frac{dy}{dx}\frac{d^3y}{dx^3}+y\frac{d^4y}{dx^4}$?

 

[Hypersphere]

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.

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.)

 

[JJacquelin]

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?

Edit:
I've noted that $\displaystyle \int\frac{d^3y}{dx^3} \, dy = \frac{d}{dx}\left[y\frac{d^3y}{dx^3}\right] = \frac{dy}{dx}\frac{d^3y}{dx^3}+y\frac{d^4y}{dx^4}$?

Sorry to say : There are some mistakes.
Did you try to test your équations with simple functions, for example y(x)=ax+b ?

 

[Mandelbroth]

Nevermind. Fixed the problem. Really bad math day.

Feel free to close the thread.