Suppose f is a continuous function on an open set Ω which has an anti-derivative on Ω. if γ, γ' are two paths in Ω with the same beginning point and the same end point, are below statement true? $$\int_{\gamma} f(z) \, dz =\int_{\gamma'} f(z) \, dz $$
My approach: I think it should be the same, since it has two curves ends in the same points.
Also by the fundamental theorem of the calculus, I know that using F as derivative of f. $$\int_{\gamma} f(z) \, dz = F(\gamma(b)) - F(\gamma(a))$$
$$\int_{\gamma'} f(z) \, dz = F(\gamma'(b)) - F(\gamma'(a))$$
But I am not sure how to approach this problem to show that these are equal. I think I need to use the fact $$\int_{\gamma} f(z) \, dz = 0$$ But not sure how to prove this.
If $\gamma$ and $\gamma'$ have the same end points, that is, $\gamma(a) = \gamma'(a)$ and $\gamma(b) = \gamma'(b)$, then obviously $F(\gamma(a)) = F(\gamma'(a))$,$F(\gamma(b)) = F(\gamma'(b))$, so $F(\gamma(b)) - F(\gamma(a)) = F(\gamma'(b)) - F(\gamma'(a))$.