Properties Of The Fundamental Theorem Of Calculus

The definite integral $\int_a^b f(x)\,dx$ is equal to $F(a)-F(b)$, where $F(x)$ is any antiderivative of $f(x)$.

If $F'(x)=f(x)$ for all x in [a, b] and $f(x)$ is continuous on [a, b], then $\int_a^b f(x)\,dx = F(b)-F(a)$.

For the Fundamental Theorem of Calculus to hold, the function $f(x)$ must be continuous on the closed interval [a, b].

Any two antiderivatives of a continuous function differ by a constant, so the constant cancels in the evaluation $F(b)-F(a)$.