TITLE: Integration with complex constant
QUESTION [5 upvotes]: This may be an overly simple question, but:
If I am integrating something like
$$\int -i\ dx\,,$$
is this the same as
$$-i \int\ dx = -i x + c, c \in \mathbb{R}\,$$
that is, the complex constant $i$ can just be treated as how one typically treats real constants in integration?
Thank you.

REPLY [2 votes]: I agree with the answer given by mrtaurho.
I just want to add that $\mathbb{C}$-linearity still holds if you are interested in finding an antiderivative of a given complex valued function, which seems to be your case to me.
This is quite obvious if you think about it: you can picture $\mathbb{C}$ simply as $\mathbb{R}^2$ endowed with a product operation that gives it a field structure. Then, functions $\mathbb{C}\rightarrow\mathbb{C}$ are just functions $\mathbb{R}^2\rightarrow\mathbb{R}^2$, and you have just traced back your problem to real numbers.
However, the problem of finding an antiderivative in several variables is not as straightforward as in one variable, and it is better understood in terms of closed and exact differential forms.
But to keep it simple and short: yes, complex constants behave in the exact same way.