Minkowski's Inequality for Integrals

Theorem
Let $f, g$ be (Darboux) integrable functions.

Let $p \in \R$ such that $p > 1$.

Then:
 * $\ds \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1/p} \le \paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p}$

Proof
Define:
 * $q = \dfrac p {p - 1}$

Then:
 * $\dfrac 1 p + \dfrac 1 q = \dfrac 1 p + \dfrac {p - 1} p = 1$

It follows that: