If one of the following conditions is true, a definite integral might not exist in a strict mathematical sense:
If the interior of the integration interval (a, b) contains poles of the integrand f(x).
If a = - ∞ or b = ∞ or both.
If f(x) changes
sign at all poles in (a, b),
the so-called infinite parts of the integral to the left and to the
right of a pole can cancel each other. In this case, use the
to find a weaker form of a definite integral called the Cauchy principal
value. For example, this integral is not defined because it has a
x = 0:
int(1/x, x = -1..1)
To compute the Cauchy principal value, call
int with the option
int(1/x, x = -1..1, PrincipalValue)
If an expression can be integrated in a strict mathematical sense, and such an integral exists, the Cauchy principal value coincides with the integral:
int(x^2, x = -1..1) = int(x^2, x = -1..1, PrincipalValue)