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:

$\displaystyle \int c f \left({x}\right) \ \mathrm dx = c \int f \left({x}\right) \ \mathrm dx$


Proof

From Linear Combination of Integrals:

$\displaystyle \int \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) \ \mathrm d x = \lambda \int f \left({x}\right) \ \mathrm d x + \mu \int g \left({x}\right) \ \mathrm d x$

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

$\blacksquare$


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