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 `PrincipalValue`

option
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
pole at `x = 0`

:

int(1/x, x = -1..1)

To compute the Cauchy principal value, call `int`

with the option `PrincipalValue`

:

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)

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