Linear Combination of Integrals

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

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

Then:
 * $$\int_a^b \left({\lambda f \left({t}\right) + \mu g \left({t}\right)}\right) dt = \lambda \int_a^b f \left({t}\right) dt + \mu \int_a^b g \left({t}\right) dt$$

Proof
Let $$F$$ and $$G$$ be primitives of $$f$$ and $$g$$ respectively on $$\left[{a \,. \, . \, b}\right]$$.

By Linear Combination of Derivatives, $$H = \lambda F + \mu G$$ is a primitive of $$\lambda f + \mu g$$ on $$\left[{a \,. \, . \, b}\right]$$.

Hence by the Fundamental Theorem of Calculus:

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