Take function \(f(x)\). From the extreme value theorem we know that:
\(\exists m \in \mathbb{R} \exists M\in \mathbb{R}\forall x\in [a,b](m<f(x)<M)\)
From continuation we know that:
\(\int_a^{x_1}f(x)dx+\int_{x_1}^{x_1+\delta x}f(x)dx=\int_a^{x_1+\delta x}f(x)dx\)
\(\int_x^{x_1+\delta x}f(x)dx=\int_a^{x_1+\delta }f(x)dx-\int_a^{x_1 }f(x)dx\)
Indefinite integrals