I've read in many places two different definitions of the second FTC, I'm interested about the general form: As presented here
If $F$ is differentiable on $[a,b]$ and the derivative $F'=f$ (say) is Riemann integrable on $[a,b]$, then: $F(b)−F(a)=\int_b^aF'(x)dx=\int_b^af(x)dx$
I would want to present a "counter-example" (in hope someone will point me some mistake):
Let $F:\Bbb R→\Bbb R$ st $F(x) = x^2\sin(1/x)$ for $x\ne0$ and $F(0)=0$
Then $F$ is differentiable on $\Bbb R$ and $F'(0) := f(0) = 0$ and $F'(x) := f(x) = 2x\sin(1/x)-\cos(1/x)$ for $x\ne0$. Then $f$ is integrable on $[0,1/(2\pi)]$. Observe that $F(1/(2\pi))-F(0) = 0-0 = 0$, but the integral from $0$ to $1/(2\pi)$ of $f$ is not equal to $0$.
