Does $\int{f(x)\cdot f'(x)}dx=\frac{f^2(x)}{2}+c$ work?

Question

$$\int{f(x) \cdot f'(x)dx}=\frac{f^2(x)}{2}+c$$ Does this rule always work, as it appears this way to me. $$\int{y\frac{dy}{dx}dx}=\int{ydy}=\frac{y^2}{2}+c$$ This is my working so far but I was wondering if this works for all functions or are there some limitations?

Answer 1

Let $u=f(x)$, then $du= f'(x)dx.$

Thus $$\int{f(x) \cdot f'(x)dx}= \int{udu} = \frac {u^2}{2}+c = \frac{f^2(x)}{2}+c $$