Linear Combination of Integrals

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

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

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

and:
 * $\displaystyle \int_a^b \left({\lambda f \left({t}\right) + \mu g \left({t}\right)}\right) \ \mathrm dt = \lambda \int_a^b f \left({t}\right) \ \mathrm dt + \mu \int_a^b g \left({t}\right) \ \mathrm dt$

Proof for Definite Integrals
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: