If an Integral Is Undefined

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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)