Primitive of Constant Multiple of Function

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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

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