이상적분improper_integral (rev. 1.3)
ex.
$\displaystyle \int_1^{\infty}\frac1xdx$
pf.
$\displaystyle p\ne 1\Rightarrow$
$\displaystyle \int_1^{\infty}\frac1xdx$
$\displaystyle =\lim_{t\to\infty}\int_1^t\frac1xdx$
$\displaystyle =\lim_{t\to\infty}\left[\ln|x|\right]_1^t$
$\displaystyle =\lim_{t\to\infty}\ln|t|$
$\displaystyle =\lim_{t\to\infty}\ln t$
$\displaystyle =\infty$ .....(1)
정리:$\displaystyle =\lim_{t\to\infty}\left[\ln|x|\right]_1^t$
$\displaystyle =\lim_{t\to\infty}\ln|t|$
$\displaystyle =\lim_{t\to\infty}\ln t$
$\displaystyle =\infty$ .....(1)
$\displaystyle \int_1^{\infty}\frac1{x^p}dx$는 $\displaystyle p>1$ 이면 수렴, $\displaystyle p\le 1$ 이면 발산.
pf.
$\displaystyle p\ne 1\Rightarrow$
$\displaystyle =\int_1^{\infty}x^{-p}dx$
$\displaystyle =\lim_{t\to\infty}\int_1^t x^{-p}dx$
$\displaystyle =\lim_{t\to\infty}\left[\frac1{1-p}x^{1-p}\right]_1^t$
$\displaystyle =\lim_{t\to\infty}\left(\frac1{1-p}t^{1-p}-\frac1{1-p}\right)$
$\displaystyle =\begin{cases}\frac1{p-1}&(p>1)\\ \infty&(p<1)\end{cases}$ (첫번째는 수렴, 두번째는 발산)
$\displaystyle =\lim_{t\to\infty}\int_1^t x^{-p}dx$
$\displaystyle =\lim_{t\to\infty}\left[\frac1{1-p}x^{1-p}\right]_1^t$
$\displaystyle =\lim_{t\to\infty}\left(\frac1{1-p}t^{1-p}-\frac1{1-p}\right)$
$\displaystyle =\begin{cases}\frac1{p-1}&(p>1)\\ \infty&(p<1)\end{cases}$ (첫번째는 수렴, 두번째는 발산)