# Linear Combination of Integrals

## 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

- 1967: Tom M. Apostol:
*Calculus Volume 1*: $\S 1.4$