I start with 1-form $\omega=f\,dx$ on $\left[0,1\right]$ where $f\left(0\right)=f\left(1\right)$ and a $g:\left[0,1\right]\to R$ with $g\left(0\right)=g\left(1\right)$ and I want to integrate $\omega-\lambda \, dx=dg$ on $\left[0,1\right]$. So I write
$\int\limits_0^1 \omega-\lambda \,dx = \int\limits_0^1 \left(f-\lambda\right)\,dx =\int\limits_0^1 f\,dx-\lambda$ and $\int\limits_0^1 \omega-\lambda \, dx =\int\limits_0^1 dg = g\left(1\right)-g\left(0\right)=0$ and find out that $$\lambda=\int\limits_0^1 f\,dx.$$
Is this correct? What would happen if I parametrized the path from $0$ to $1$ another way? Is $\int\limits_0^1 dg=g\left(1\right)-g\left(0\right)$ legal - I'm using dg as a differential form and maybe I'm supposed to check some precondition?