Contradictory results of the integral of an odd function

Question

I got two contradictory results for the integral of an odd function. I believe I made some kind of mistake but I can't figure it out.

$f(x)$ is an odd function, which means $f(-x) = -f(x)$ for any $x$. $F(x)$ is an integral of $f(x)$. So $F(x) = \int f(x) dx$.

But the integral can be calculated like this.

$$ \int f(x) dx = \int f(-y) d(-y) \,\,(y=-x) \\ = -\int f(y) \frac{d(-y)}{dy} dy \\ = \int f(y) dy \\ =F(y) \\ =F(-x)$$

$\int f(x) dx = F(x)$ and $\int f(x) dx = F(-x)$. It's quite weird. Why is this happening?

Answer 1

HINT

Consider that

$$f(x)=x \implies \int x dx = \frac12 x^2 +c$$

and also note that

$$f(x)=f(-x)\implies f'(x)=-1\cdot f'(-x)$$