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:

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


Proof

From Linear Combination of Integrals:

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

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

$\blacksquare$


Sources