# What kind of integral is the one in Laplace transform?

In a book about circuits I found a chapter where it is defined the Laplace transform of a function $$f:\mathbb{R}\rightarrow\mathbb{R}$$ as $$L[f(t)]=\int_0^{+\infty}f(t)e^{-st}dt$$ where $$s\in\mathbb{C}$$. This is the first time I find an integral where the function to integrate is a complex function and I would like to know what kind of integral this is (how it is defined) because I could not ind it anywhere else.

There are two main cases ; if one wants to integrate a function $$f:\mathbb{R} \to \mathbb{C}$$, then the problem can be reduced to integrating two functions $$x \mapsto \Re(f(x))$$ and $$x \mapsto \Im(f(x))$$. Indeed, we have the following equality :
$$\int_I f(t) dt = \int_I \Re(f(t)) dt + i \int_I \Im(f(t)) dt.$$
However, do not use that equality everytime. Some complex valued functions (and a lot of them are usual functions) can be integrated using the same rules as real ones. For example, the antiderivative of $$x \mapsto e^{ix}$$ is $$x \mapsto \frac{e^{ix}}{i}$$, just as one would have $$x \mapsto \frac{e^{ax}}a, a \in \mathbb{R}$$ being the antiderivative of $$x \mapsto e^{ax}$$, we just had to consider $$i$$ as a number and apply the usual rule.