Let $df = \frac{\partial f}{\partial x_1 } dx_1 + \frac{\partial f }{\partial x_2 } dx_2$ be a $1-form.$
I know that that the line integral (along a curve $\gamma:[0,t] \rightarrow \mathbb{R}^2$) is defined as
$$\int_{\gamma} df = \int_{0}^{t} df(\gamma(t)) \gamma'(t) dt,$$
but I don't know what exactly this last term means, is it just
$$\int_{\gamma} df = \int_{0}^{t} \frac{\partial f(\gamma(t))}{\partial x_1} \gamma_1'(t) + \frac{\partial f( \gamma(t))}{\partial x_2} \gamma_2'(t) dt?$$
That is what I would suspect, but I don't know how it exactly follows from the definition?