Change of variables is an extremely powerful method for performing integrals not only analytically but also numerically. A change of variables can considerably improve the accuracy of regular-interval techniques for functions with rapid variations in particular regions of the integration domain and can allow one to perform integrals which would otherwise be impossible, such as for functions with integrable singularities or integrals over infinite domains.
Change of variables begins with the familiar analytic formula,
where r(x) is any function which maps the unit interval 
to the interval 
. The most trivial example of
such a map is the linear map
which transforms integration over any interval to integration over the
unit interval. Figure 4(a) illustrates such a map along
with the distribution of resulting sample points 
.
Figure 4: Mapping of regular sampling of unit interval under change of
variables: (a) linear map, (b) map to semi-infinite interval, (c) map
to semi-infinite interval with concentration of points at origin.
Maps which transform integration over an infinite interval to integration over a finite interval represent a far more important example. The map
for instance, makes possible the evaluation of integrals over the
interval 
with the techniques discussed above. As 
, 
, and this map spaces the sample
points regularly near the origin. However, As 
, however, the sample points eventually tend toward infinity.
(See Figure 4(b).) Under this map, the integral
becomes
which Figure 5 illustrates to be much more manageable. In
practice, such changes of variables require care to ensure that the
proper limiting values for f(r(x)) (dr/dx) are used at the endpoints
where the change of variables factor 
becomes
singular.
Figure: Integrand of 
: (a) direct
integration, (b) under
change of variables in Eq. (21).
Finally, as an example of a problem with an integrable singularity consider the integral
As 
, the integrand approaches 
and
diverges toward infinity. (See Figure 6(a).)
Heuristically, the solution is to concentrate sample points
near the singularity at the origin. A map which both extends the
range of integration to infinity and concentrates points near the
origin is
(See Figure 4(c).) Figure 6(b) shows that the resulting integrand f(r(x)) (dr(x)/dx) is now well behaved and quite suited to regular interval techniques.
Figure: Integrand of 
: (a) direct
integration, (b) under
change of variables in Eq. (22).
