Can anyone help me to prove that these integral inequalities hold?
Here $x$ is a real value:
$$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$
Here $z$ is a complex value:
$$ \left| \int_C^ \ f(z) dz \right| \leq \int_C^\ |f(z)| |dz| $$