Linear Combination of Integrals

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Theorem

Let $f$ and $g$ be real functions which are integrable on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $\lambda$ and $\mu$ be real numbers.


Then the following results hold:


Indefinite Integrals

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


Definite Integrals

$\displaystyle \int_a^b \left({\lambda f \left({t}\right) + \mu g \left({t}\right)}\right) \rd t = \lambda \int_a^b f \left({t}\right) \rd t + \mu \int_a^b g \left({t}\right) \rd t$


Sources