Answer
See below.
Work Step by Step
Let $F(x)$ be an antiderivative of $f(x),\qquad F'(x)=f(x)$
Let $G(x)$ be an antiderivative of $g(x),\qquad G'(x)=g(x)$.
By the rule for sums of derivatives
$[ F(x)+G(x) ]'= F'(x)+G'(x)$
so, $F(x)+G(x)$ is an antiderivative of $f(x)+g(x)$ .
We write this as
$\displaystyle \int\lceil f(x)+g(x)]dx=F(x)+G(x)+C$
writing the constant C as a sum of two other constants,
$=(F(x)+C_{1})+(G(x)+C_{2})$
where we defined $F$ and $G,$
$=\displaystyle \int f(x)dx+\int g(x)dx$
(The integral of a sum is the sum of integrals).
