Linear Combination of Integrals/Definite/Proof 1

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: 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 by the Fundamental Theorem of Calculus: