An improper integral is one where either of the following holds
one of the integrals is
or
or the limits areand
and
are all improper integrals.
the integrand (the function being integrated) includes a term evaluated at one or both limits which takes the form
and
are all improper integrals. The first of these integrands,
includes the factor
which tends toastends toand
which tends to 0 astends to infinity . The second includes the factors
and
which tend toand 0 respectively astends to
The third includes the termsand
which tend to 0 andastends to 0.
Improper integrands can often be evaluated because the integrand tends to 0 at the troublesome limit, or if the integrand is of the form
at one or both limits, one factor tends to 0 faster than the other tends to infinity. This is true for the improper integral
tends to zero faster than any power oftends to infinity,
so we may write
Having the integrand tend to zero at the limits is not sufficient for the integral to be able to be evaluated. The integrand must tend to 0 fast enough. The integral
is not defined because
does not tend to 0 fast enough. In fact,