# Definite Integral of Constant Multiple of Real Function/Proof 1

## Theorem

Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.

Let $c \in \R$ be a real number.

Then:

$\displaystyle \int_a^b c \, \map f x \rd x = c \int_a^b \map f x \rd x$

## Proof

Let $F$ be a primitive of $f$ on $\closedint a b$.

By Primitive of Constant Multiple of Function, $H = c F$ is a primitive of $c f$ on $\closedint a b$.

Hence by the Fundamental Theorem of Calculus:

 $\displaystyle \int_a^b c \map f x \rd x$ $=$ $\displaystyle \bigintlimits {c \map F x} a b$ $\displaystyle$ $=$ $\displaystyle c \bigintlimits {\map F x} a b$ $\displaystyle$ $=$ $\displaystyle c \int_a^b \map f x \rd x$

$\blacksquare$