Improper Integrals of Real Functions :
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Note: It is important to note the following:

  1. If the improper integral MATH converges, then the Cauchy principal value MATH exists. Further, MATH.

  2. When the Cauchy principal value MATH exists, it is not always true that MATH converges.
    For example, MATH exists, but MATH fails to exist.

  3. If $f(x)$ is an even function on $\QTR{Bbb}{R}$ (That is, $f(-x) = f(x)$ for all $x \in \QTR{Bbb}{R}$) and if the Cauchy principal value MATH exists, then the improper integral MATH exists and equal to MATH.

Note: If $f(x)$ is not an even function, then it is always a good practice to write $P.V.$ explicitly before the integral symbol to denote the Cauchy principal value of the improper integral.

 
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