The FTC corollary given in wikipedia is
if $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$, then
$$\int_a^bf(x)dx = F(b) – F(a).$$
My book gives a different version.
If $f$ is continuously differentiable on $[a, b]$, then
$$\int_a^b f'(x)dx = f(b)-f(a).$$
Are these two versions equivalent? If so, how can I see that they are?
Best Answer
All that FTC states is, $$\int_a^b \textrm{some function}=\left[\textrm{it's antiderivative}\right]_a^b$$ holds whenever $\textrm{some function}$ is continuous on the closed interval.
It's your choice to call the function $f$, with its antiderivative being $F$, in which case it'd require $f$ to be continuous on $[a,b]$. Or you may call the function $f'$ with its antiderivative being simply $f$, in which case it'd require $f'$ to be continuous over $[a,b]$.
Hope this helps. Ask anything if not clear :)