I try to show that the following analogous result for $\textbf{closed curve}$ :
Let $\gamma_0, \gamma_1$ be continuous paths in a domain $D \subseteq \mathbb{C}$ and $\omega$ be a closed form on $D$. If $\gamma_0$ is homotopic to $\gamma_1$ with their end points fixed (so they are not closed curve), then $\int_{\gamma_0} \omega = \int_{\gamma_1} \omega$.
This is what I try so far :
Let $\gamma_0, \gamma_1 : [0,1] \rightarrow D$ with $\gamma_i(0) = \gamma_i(1)$ for $i = 1,2$. Let $H$ be a homotopy between closed curve $\gamma_0, \gamma_1$ ($H : [0,1] \times [0,1] \rightarrow D$). Let $f$ be a primitive of $\omega$ along $H$ (it existence is confirmed by a theorem). Then $$\int_{\gamma_0} \omega = f(1,0) - f(0,0)$$ because $f(t,0)$ is a primitive for $\omega$ along $\gamma_0$. Also $$\int_{\gamma_1} \omega = f(1,1) - f(0,1).$$ This method works for any non-closed curves since their end points are fixed and equal such that $f(1,0) = f(1,1), f(0,0) = f(0,1)$. But this fact is not true for closed curves since each of them might do not share any points in common.
Here are some $\textbf{definitions}$ involved in solving this problem :
$\textbf{Definition 1}$ A domain $D \subseteq \mathbb{C}$ is an open connected set.
$\textbf{Definition 2}$ A differential form $\omega$ is exact on $D$ if there exists a $C^1$-function $F$ ($F$ has continuous first derivative) on D such that $\omega = dF$. $\omega$ is closed form if for each $c\in D$, there exists a neighborhood $U$ of $c$ such that the restriction $\omega|_U$ is exact.
For definitions concerning homotopy between curves, I can provide you if it is needed (the definition is quite long). Also, as a reference, all information can be found in Complex Analysis in spirit of Lipman Bers (2nd edition)
Thank you in advanced for help.
$\textbf{I come up with some ideas as follows}$ Between the two closed path I draw two lines connecting these two paths. Then I have two news paths (Actually, one of them is the old path) with common fixed starting and ending points. Then I apply the theorem for non-closed paths. With some adjustment, I think that can obtain the result. Is this idea possible ? Any comment,please ?