Editing Improper integral (section)

==Cauchy principal value==
{{main article|Cauchy principal value}}
Consider the difference in values of two limits:

:<math>\lim_{a\to 0^+}\left(\int_{-1}^{-a}\frac{dx}{x}+\int_a^1\frac{dx}{x}\right)=0,</math>

:<math>\lim_{a\to 0^+}\left(\int_{-1}^{-a}\frac{dx}{x}+\int_{2a}^1\frac{dx}{x}\right)=-\ln 2.</math>

The former is the Cauchy principal value of the otherwise ill-defined expression

:<math>\int_{-1}^1\frac{dx}{x}{\  }
\left(\mbox{which}\  \mbox{gives}\  -\infty+\infty\right).</math>

Similarly, we have

:<math>\lim_{a\to\infty}\int_{-a}^a\frac{2x\,dx}{x^2+1}=0,</math>

but

:<math>\lim_{a\to\infty}\int_{-2a}^a\frac{2x\,dx}{x^2+1}=-\ln 4.</math>

The former is the principal value of the otherwise ill-defined expression

:<math>\int_{-\infty}^\infty\frac{2x \, dx}{x^2+1}{\  }
\left(\mbox{which}\  \mbox{gives}\  -\infty+\infty\right).</math>

All of the above limits are cases of the [[indeterminate form]] <math>\infty - \infty</math>.

These [[pathological (mathematics)|pathologies]] do not affect "Lebesgue-integrable" functions, that is, functions the integrals of whose [[absolute value]]s are finite.