Primitive of Pointwise Sum of Functions/Examples/u+v-w

Examples of Use of Primitive of Pointwise Sum of Functions

Let $u$, $v$ and $w$ be real functions of $x$ which are integrable.

Then:

$\ds \int \paren {u + v - w} \rd x = \int u \rd x + \int v \rd x - \int w \rd x$

Proof

This is an instance of Primitive of Pointwise Sum of Functions:

$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$

where $u = f_1$, $v = f_2$ and $w = f_3$.

$\blacksquare$

Sources

• 1960: Margaret M. Gow: A Course in Pure Mathematics ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: General Rules: $\text I$.