I was thinking about one question from the book and got quite a bit of confusion and had no idea where to start. The text is Henri Cartan's "Elementary Theory of Analytic Functions of One or Several Complex Variables". I apologize in advance if the terminology used in this book is not standard or anything, and also about this long post. But I guess anyone familiar with complex analysis using differential forms could think this is very easy to read.
The question is stated as follows:
Let $\gamma$ be a continuous path (not necessarily piecewise differentiable). Show that $$\int_{\gamma} (\omega_1+\omega_2) = \int_{\gamma} \omega_1 + \int_{\gamma} \omega_2$$, where $\gamma_1$ and $\gamma_2$ are closed differential forms.
Possible useful definitions in my opinion (not sure):
If $\gamma_0$ and $\gamma_1$ are two homotopic paths of an open set $D$ of the complex plane, with fixed end points, then $$\int_{\gamma_0} \omega = \int_{\gamma_1} \omega$$ for any closed form $\omega$ in $D$.
Let $\gamma: [a,b] \mapsto D $ be a path contained in an open set D, and let $\omega$ be a closed differential form in $D$. A continuous function $f(t)$ ($t$ describing $[a,b]$) is called a primitive of $\omega$ along $\gamma$ if it satisfies the following condition: For any $\tau \in [a,b]$ there exists primitive $F$ of $\omega$ in a neighborhood of the point $\gamma (\tau) \in D$ such that $$F(\gamma (t)) = f(t)$$ for $t$ near enough to $\tau$.
Another definition follows from #2 above. This definition is probably the most relevant for this question. For a continuous path $\gamma$ (i.e. without the assumption of differentiability of $\gamma$), if $f$ is a primitive along $\gamma$, then we can define the following: $$\int_{\gamma} = f(b)-f(a)$$.
The primitive $f$ in #2 always exists, and up to a constant.