Linear Combination of Integrals/Indefinite
< Linear Combination of Integrals(Redirected from Linear Combination of Primitives)
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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: $\S 1.4$