Main theorem of calculus/Riemann/Newton-Leibniz-formula/Fact/Proof

Proof

Due to

the integral exists. With the integral function

${}G(x):=\int _{a}^{x}f(t)\,dt\,,$

we have the relation

${}\int _{a}^{b}f(t)\,dt=G(b)=G(b)-G(a)\,.$

Because of

the function ${}G$ is differentiable and

${}G'(x)=f(x)\,$

holds. Hence ${}G$ is a primitive function for ${}f$. Due to

we have ${}F(x)=G(x)+c$. Therefore,

${}\int _{a}^{b}f(t)\,dt=G(b)-G(a)=F(b)-c-F(a)+c=F(b)-F(a)\,.$

To fact