Suppose we have the integral



We can make the substitution , which is undoubtedly valid because the range of the substitution is the same as the domain of the integrand .

But what about an integrand whose domain is all real numbers? For example:



Obviously we can integrate this without a substitution to get but suppose we wanted to use . We would end up with , which is still correct.

But is the substitution valid? Since the range of the substitution is still , which doesn't cover the domain of the integrand .