What is the formal definition of "orientation" of a curve? Or is this a topology issue?

Let be a smooth function. And is continous on where , i.e. the image. And is another such function. With the property that and have the same orientation. Then show that .

Basically, what I am trying to show that if we want to integrate a complex function along some path it makes no difference how we parametrize it (as long the orientation is the same).