B Rule to integrate a function with respect to its derivative

[Bill_Nye_Fan]

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 :)

 

[fresh_42]

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 $f'(x)$.

 

[andrewkirk]

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 $f$.

 

[Bill_Nye_Fan]

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.