TITLE: Calculate $\int f(x) f''(x)dx$
QUESTION [5 upvotes]: I have a curiosity. If
$\int f(x) f'(x)dx=\int f(x) df(x)=\frac{\left(f(x)\right)^{2}}{2}+C$ 
what is the result of:
$\int f(x) f''(x)dx$

REPLY [2 votes]: Using by parts $\int f(x)f''(x)dx= f(x)f'(x)-\int f'^2(x)dx$.

REPLY [2 votes]: $$
f(x)f'(x)-\int f'(x)^2 \, dx \ ?
$$