查看“︁微积分学/定积分/反常积分”︁的源代码

{{noteTA|zh-hans:反常;zh-hant:瑕}}
{{wikipedia|反常積分}}
==例題==
<math>
\begin{align}&
1.\;\int_0^3\frac1{x^2-6x+5}dx
\\&
=\int_0^3\frac1{(x-1)(x-5)}dx
\\&
=\int_0^3\frac A{(x-1)}+\frac B{(x-5)}dx
\\&
\qquad A=\frac{-1}4\qquad B=\frac14
\\&
=\int_0^3\frac{-1}4\frac1{(x-1)}+\frac14\frac{(x-5)}dx
\\&
=\frac{-1}4\int_0^3\frac1{(x-1)}dx+\frac14\int_0^3\frac1{(x-5)}dx
\\&
=\frac{-1}4\left(\lim_{b\to1^-}\int_0^b\frac1{(x-1)}dx+\lim_{b\to1^+}\int_b^3\frac1{(x-1)}dx\right)+\frac14\bigg[\ln(x-5)\bigg]_0^3
\\&
=\frac{-1}4\left(\lim_{b\to1^-}\bigg[\ln(x-1)\bigg]_0^b+\lim_{b\to1^+}\bigg[\ln(x-1)\bigg]_b^3\right)+\frac14\ln\frac{-2}{-5}
\\&
=\frac{-1}4\left(\lim_{b\to1^-}\ln\frac{b-1}{-1}+\lim_{b\to1^+}\ln\frac2{b-1}\right)+\frac14\ln\frac25
\\&
=\frac{-1}4\left(\ln0^++\ln1+\ln2-\ln0^+\right)+\frac14\ln\frac25
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=\frac{-1}4\left(\infty-\infty+\ln2\right)+\frac14\ln\frac25
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=\text{undefined}
\\&
\\&
2.\;\int_{-\infty}^\infty\frac1{1+x^2}dx
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\quad\tan\frac\pi2=\infty\qquad\tan\frac{-\pi}2=-\infty
\\&
=\Big[\tan^{-1}x\Big]_{-\infty}^\infty=\frac1\pi+\frac1\pi=\pi
\end{align}
</math>