Anti-derivative of an analytical expression

Question

I have a function $g : \mathbb{R}^d \rightarrow \mathbb{R}$ and I want to find the anti-derivative of the following expression: $$ \partial_{x_1} g(x) + a $$ where the differentiation is taken with repect to $x_1$ in $\partial_{x_1} g(x)$ and $a \in \mathbb{R}$. For the second term, it should be $x_1$ but for the first one can I simply say $g(x)$? as obviously $g(x_1)$ is not well-defined?

Answer 1

An anti-derivative is given by

$$g(x)+ax_1+\varphi(x_2,...,x_n).$$

$\varphi$ is a (unknown) function, which does not depend on $x_1.$