integral after substituion of $x\to-y$

Question

I just cant spot the mistake in my calculation. As I said in the title, I used the substituion $x\rightarrow-y$, $dx\rightarrow-dy$ and $a<b$ \begin{align} F(b)-F(a)&=\big[F(x)\big]_a^b=\int_a^bf(x)dx\\&=\int_{-a}^{-b}-f(-y)dy\\&=\big[-F(-y)\big]_{-a}^{-b}\\&=-F(b)+F(a) \end{align} which clearly isnt true since $$ F(b)-F(a)\neq-F(b)+F(a) $$ Can someone please spot the mistake for me ? I just keep looking at it but cant find the mistake

Answer 1

If $F(x)$ is a primitive of $f$, then a primitive of $-f(-y)$ is $F(-y)$. You have no reason to assume that $-F(-y)$ is a primitive of $-f(-y)$, which is what you did.