Primitive of Pointwise Sum of Functions/Proof 2

Theorem

Let $f_1, f_2, \ldots, f_n$ be real functions which are integrable.

Then:

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

Proof

From Sum Rule for Derivatives:

$\ds \map {\dfrac \d {\d x} } {\sum_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \map {\dfrac \d {\d x} } {\map {f_i} x}$

The result follows by definition of primitive.

$\blacksquare$

Sources

• 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Three rules for integration: $\text {II}$