Primitive of Constant Multiple of Function

Theorem

Let $f$ be a real function which is integrable.

Let $c$ be a constant.


Then:

$\ds \int c \map f x \rd x = c \int \map f x \rd x$


Proof 1

From Linear Combination of Primitives:

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

The result follows by setting $\lambda = c$ and $\mu = 0$.

$\blacksquare$


Proof 2

From Derivative of Constant Multiple:

$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$

The result follows from the definition of primitive.


Examples

Primitive of $2 \cos x$

$\ds \int 2 \cos x \rd x = 2 \sin x + C$


Sources

  • 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: General Rules: $\text {II}$.
  • 1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.2$
  • 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (next): Appendix $\text I$: Table of Indefinite Integrals $1$.
  • 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $25$: Fundamental Integration Formulas: $3$.
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.2.$
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