In many instances it is either a matter of convenience or of necessity to make a substitution, or change of variable, in a definite integral in order to evaluate it. If f is continuous on [a, b], x = g(u) has a continuous derivative, and dx = g′(u) du, then
f(x) dx = f(g(u)) g′(u) du,(1)
where the u-limits of integration c and d are defined by a = g(c) and b = g(d). There are three things ...