Substitution in Integrals

In chapter 12, we saw how a great many integrals, of the type ∫f (Ax + B) dx, may be derived from the integrals in the standard table by means of a simple rule. In fact the rule given in Chapter 12 is a special case of a more general rule for substituting in integrals. In the method of substitution, we try to reduce a given integral to one of the standard types by picking out a likely expression in x which we call u(x), and then expressing the whole integral in terms of u. In this case, of course, we must also express dx in terms of du; but the rule for this is quite easy since in differential notation: dx = dx/du.du, and we are allowed to make this substitution under the integral sign (see class discussion exercise 2) to give the rule: ∫ ... d x = ∫ ... d x d u d u $$ \boxed{\int {...dx = \int {...\frac{{dx}}{{du}}} } du} $$