It is usually the case that an integral f f (x ) dx cannot be integrated as it stands because its integrand f (x ) is more complicated than any of those listed in the short list of integrals given in Section 33. In many cases, however, if the
variable x is replaced by a suitable function of x , by say, u/U (x ), the integral in terms of u is simpler and can be evaluated by direct appeal to
the list of integrals. The result of this substitution is to give a result in terms
of u , but by writing u/U (x) we return to the original variable x and so obtain the required result. For obvious reasons, this method of evaluating
integrals (antiderivatives) is called integration by substitution.