Since $\int \frac{df(x)}{dx} dx = f(x)+c$, can we cancel $dx$ and write $\int df(x) = f(x)+c$?

Question

Consider $$\int \frac{df(x)}{dx}dx$$

It's the same as $f(x)+c$ because we just differentiate and then integrate f(x).

But on the other hand, cancel out $dx$ gives $\int df(x)$

So $$f(x)+c=\int df(x)$$

Is this true? If so, what's $\int df(x)$?

Answer 1

set $y=f(x)$, we have $dy=f'(x)dx$, $$\int f'(x)dx=\int f'(x)\frac{dy}{f'(x)}=\int dy=y+c=f(x)+c$$ This is just the Variable transformation in integration.