Linear Combination of Integrals/Indefinite

Theorem
Let $f$ and $g$ be real functions which are integrable on the closed interval $\closedint a b$.

Let $\lambda$ and $\mu$ be real numbers.

Then:
 * $\displaystyle \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$

Proof
Let $F$ and $G$ be primitives of $f$ and $g$ respectively on $\closedint a b$.

By Linear Combination of Derivatives, $H = \lambda F + \mu G$ is a primitive of $\lambda f + \mu g$ on $\closedint a b$.

Hence: