Linear Combination of Integrals

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 the following results hold:

Indefinite Integrals

$\ds \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$

Definite Integrals

$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$

Sources

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• 1967: Tom M. Apostol: Calculus Volume 1: $\S 1.4$