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:

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

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:

\(\ds \int \paren {\lambda \map f t + \mu \map g t} \rd t\)

\(=\)

\(\ds \lambda \map F t + \mu \map G t\)

\(\ds \)

\(=\)

\(\ds \lambda \int \map f t \rd t + \mu \int \map g t \rd t\)

$\blacksquare$

Sources

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