Linear Combination of Integrals/Indefinite

Theorem
Let $f$ and $g$ be real functions which are integrable on the closed interval $\left[{a \,.\,.\, b}\right]$.

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

Then:
 * $\displaystyle \int \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) \rd x = \lambda \int f \left({x}\right) \rd x + \mu \int g \left({x}\right) \rd x$

Proof
Let $F$ and $G$ be primitives of $f$ and $g$ respectively on $\left[{a \,.\,.\, b}\right]$.

By Linear Combination of Derivatives, $H = \lambda F + \mu G$ is a primitive of $\lambda f + \mu g$ on $\left[{a \,.\,.\, b}\right]$.

Hence: