Primitive of Pointwise Sum of Functions

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

Then:


 * $\displaystyle \int \left({f_1 \pm f_2 \pm \, \cdots \pm f_n}\right) \left({x}\right) \ \mathrm d x = \int f_1 \left({x}\right) \ \mathrm d x \pm \int f_2 \left({x}\right) \ \mathrm d x \pm \, \cdots \pm \int f_n \left({x}\right) \ \mathrm d x$

Proof
Proof by induction:

For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \int \left({f_1 \pm f_2 \pm \, \cdots \pm f_n}\right) \left({x}\right) \ \mathrm d x = \int f_1 \left({x}\right) \ \mathrm d x \pm \int f_2 \left({x}\right) \ \mathrm d x \pm \, \cdots \pm \int f_n \left({x}\right) \ \mathrm d x$

$P \left({1}\right)$ is true, as this just says:
 * $\displaystyle \int f_1 \left({x}\right) \ \mathrm d x = \int f_1 \left({x}\right) \ \mathrm d x$

Basis for the Induction
$P \left({2}\right)$ is the case:

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $\displaystyle \int \left({f_1 \pm f_2 \pm \, \cdots \pm f_k}\right) \left({x}\right) \ \mathrm d x = \int f_1 \left({x}\right) \ \mathrm d x \pm \int f_2 \left({x}\right) \ \mathrm d x \pm \, \cdots \pm \int f_k \left({x}\right) \ \mathrm d x$

Then we need to show:
 * $\displaystyle \int \left({f_1 \pm f_2 \pm \, \cdots \pm f_{k+1}}\right) \left({x}\right) \ \mathrm d x = \int f_1 \left({x}\right) \ \mathrm d x \pm \int f_2 \left({x}\right) \ \mathrm d x \pm \, \cdots \pm \int f_{k+1} \left({x}\right) \ \mathrm d x$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \N: \displaystyle \int \left({f_1 \pm f_2 \pm \, \cdots \pm f_n}\right) \left({x}\right) \ \mathrm d x = \int f_1 \left({x}\right) \ \mathrm d x \pm \int f_2 \left({x}\right) \ \mathrm d x \pm \, \cdots \pm \int f_n \left({x}\right) \ \mathrm d x$