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:

 $\displaystyle \int \paren {\lambda \map f t + \mu \map g t} \rd t$ $=$ $\displaystyle \lambda \map F t + \mu \map G t$ $\displaystyle$ $=$ $\displaystyle \lambda \int \map f t \rd t + \mu \int \map g t \rd t$

$\blacksquare$