This is a very basic question about how integrals distribute over multiple variables. Suppose one has functions $f(x_1)$, $g(x_2)$, and $h(x_3)$ with antiderivatives $F(x_1)$, $G(x_2)$, and $H(x_3)$. Which of the following two expressions, if either, concerning $\iiint \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3 \left( f(x_1) + g(x_2) + h(x_3)\right)$ is correct?
$\begin{align} \text{(1) } &\iiint \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3 \left( f(x_1) + g(x_2) + h(x_3)\right) \\ = &\iiint \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3 f(x_1) + \iiint \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3g(x_2) + \iiint \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3h(x_3) \\ = &F(x_1) \cdot x_2 \cdot x_3 + G(x_2) \cdot x_1 \cdot x_3 + H(x_3) \cdot x_1 \cdot x_2 + C \end{align}$
\begin{align} \text{(2) } &\iiint \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3 \left( f(x_1) + g(x_2) + h(x_3)\right) \\ = &\int \mathrm{d}x_1 f(x_1) + \int \mathrm{d}x_2 g(x_2) + \int \mathrm{d}x_3 h(x_3) \\ = &F(x_1) + G(x_2) + H(x_3) + C' \end{align}