If I understood well the change of the limits of the integral when we make a substitution, it must looks like this :

Say we have Let , then . So we have . I changed the limits of the integral of and to and because when , , etc. But now comes the step I don't understand : . Why the limits change again? It seems totally worthless to bother to change the limits of the integral if at last we change them as they were! I don't understand the why of we change the limits in the last step. (I understand why in the first step, it's because of the chain rule).