Primitive of Pointwise Sum of Functions/Examples/f+g

Examples of Use of Primitive of Pointwise Sum of Functions

Let $f$ and $g$ be real functions of $x$ which are integrable.

Then:

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

Proof

This is an instance of Primitive of Pointwise Sum of Functions:

$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$

where $f = f_1$ and $g = f_2$.

$\blacksquare$

Sources

• 1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
• 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Appendix $\text I$: Table of Indefinite Integrals $2$.
• 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (next): Chapter $25$: Fundamental Integration Formulas: $2$.

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