{"id":8545,"date":"2023-05-01T15:42:03","date_gmt":"2023-05-01T06:42:03","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=8545"},"modified":"2023-05-04T23:30:45","modified_gmt":"2023-05-04T14:30:45","slug":"tests-for-improper-integrals","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/tests-for-improper-integrals\/","title":{"rendered":"\uc774\uc0c1\uc801\ubd84\uc758 \uc815\uc758\uc640 \uc218\ub834 \ud310\uc815\ubc95"},"content":{"rendered":"<p>\uc774 \uae00\uc5d0\uc11c\ub294 \ubcc0\uc218\uac00 \ud558\ub098\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\uc758 \uc774\uc0c1\uc801\ubd84\uc744 \uc815\uc758\ud558\uace0, \uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834 \ud310\uc815\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub610\ud55c \uc774\uc0c1\uc801\ubd84\uc744 \ud65c\uc6a9\ud558\ub294 \uc608\ub85c\uc11c \uac10\ub9c8 \ud568\uc218\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#introduction\">\ub4e4\uc5b4\uac00\uae30<\/a><\/li>\n<li><a href=\"#infiniteinterval\">\uae38\uc774\uac00 \ubb34\ud55c\uc778 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\ub294 \uc774\uc0c1\uc801\ubd84<\/a><\/li>\n<li><a href=\"#unboundedfunction\">\uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\uc758 \uc774\uc0c1\uc801\ubd84<\/a><\/li>\n<li><a href=\"#convergencetest1\">\uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834 \ud310\uc815\ubc95 (\uc801\ubd84 \uad6c\uac04\uc758 \uae38\uc774\uac00 \ubb34\ud55c\uc778 \uacbd\uc6b0)<\/a><\/li>\n<li><a href=\"#convergencetest2\">\uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834 \ud310\uc815\ubc95 (\ud568\uc218\uac00 \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0)<\/a><\/li>\n<li><a href=\"#gammafunction\">\uc774\uc0c1\uc801\ubd84\uc744 \ud65c\uc6a9\ud558\ub294 \uc608: \uac10\ub9c8 \ud568\uc218<\/a><\/li>\n<li><a href=\"#outtro\">\ub9fa\uc74c\ub9d0<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>\uc815\uc801\ubd84 (<a href=\"\/blog\/invitation-to-calculus\/definite-integrals\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac (<a href=\"\/blog\/invitation-to-calculus\/fundamental-theorem-of-calculus\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"introduction\"><\/a><\/p>\n<h3>\ub4e4\uc5b4\uac00\uae30<\/h3>\n<p>\ub9ac\ub9cc \uc801\ubd84\uc740 \uae38\uc774\uac00 \uc720\ud55c\uc778 \ub2eb\ud78c \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ub41c\ub2e4. \uadf8\ub7ec\ub098 \uc801\ubd84 \uad6c\uac04\uc758 \uae38\uc774\uac00 \uc720\ud55c\uc774 \uc544\ub2c8\uac70\ub098 \uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\ub97c \uc801\ubd84\ud574\uc57c \ud560 \ub54c\uac00 \uc788\ub294\ub370, \uc774\ub54c \uc0ac\uc6a9\ub418\ub294 \uac83\uc774 \uc774\uc0c1\uc801\ubd84\uc774\ub2e4.<\/p>\n<p>\uc774\uc0c1\uc801\ubd84\uc744 \uc815\uc758\ud558\uae30 \uc704\ud558\uc5ec \uc6b0\uc120 \u2018\uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\u2019(locally integrable)\ub77c\ub294 \uac1c\ub150\uc744 \uc815\uc758\ud574\uc57c \ud55c\ub2e4. \ud568\uc218 \\(f\\)\uac00 \uc9d1\ud569 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(I\\)\ub294 \ub2eb\ud78c \uad6c\uac04\uc77c \uc218\ub3c4 \uc788\uace0 \uc5f4\ub9b0 \uad6c\uac04\uc77c \uc218\ub3c4 \uc788\uc73c\uba70, \uae38\uc774\uac00 \uc720\ud55c\uc77c \uc218\ub3c4 \uc788\uace0 \ubb34\ud55c\uc77c \uc218\ub3c4 \uc788\ub2e4. \ub610\ud55c \\(I\\)\ub294 \uc5ec\ub7ec \uac1c\uc758 \uad6c\uac04\uc758 \ud569\uc9d1\ud569\uc77c \uc218\ub3c4 \uc788\ub2e4. \ub9cc\uc57d \\([a,\\,\\,b]\\)\uac00 \\(I\\)\uc758 \ubd80\ubd84\uad6c\uac04\uc774\uace0 \uae38\uc774\uac00 \uc720\ud55c\uc77c \ub54c\ub9c8\ub2e4 \\(f\\)\uac00 \\([a,\\,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uba74, \u201c\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4<\/span>\u201d\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \ub450 \uac00\uc9c0 \ud615\ud0dc\uc758 \uc774\uc0c1\uc801\ubd84\uc744 \uc815\uc758\ud558\uace0, \uc774\ub4e4\uc758 \uc218\ub834\uc744 \ud310\uc815\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"infiniteinterval\"><\/a><\/p>\n<h3>\uae38\uc774\uac00 \ubb34\ud55c\uc778 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\ub294 \uc774\uc0c1\uc801\ubd84<\/h3>\n<p>\\(a\\)\uac00 \uc2e4\uc218\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\(f\\)\uac00 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \uadf9\ud55c<br \/>\n\\[\\lim_{b\\rightarrow\\infty} \\int_{a}^{b} f(x) dx\\tag{1}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uba74, \u201c\uad6c\uac04 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4<\/span>\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uadf8 \uadf9\ud55c\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\int_{a}^{\\infty} f(x) dx \\tag{2}\\]<br \/>\n\uc774 \uadf9\ud55c\uc758 \uac12\uc744 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc774\uc0c1\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div>\n<a href=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01.png\" data-rel=\"penci-gallery-image-content\" ><img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01.png\" alt=\"\" width=\"373\" height=\"204\" class=\"aligncenter size-full wp-image-8698\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01.png 2239w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-300x164.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-1024x559.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-768x419.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-1536x838.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-2048x1118.png 2048w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-1920x1048.png 1920w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-1170x639.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-585x319.png 585w\" sizes=\"(max-width: 373px) 100vw, 373px\" \/><\/a>\n<\/div>\n<p><!--\n\n\n<p>\n<a href=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01.png\" data-rel=\"penci-gallery-image-content\" ><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01.png\" alt=\"\" width=\"2239\" height=\"1222\" class=\"aligncenter size-full wp-image-8698\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01.png 2239w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-300x164.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-1024x559.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-768x419.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-1536x838.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-2048x1118.png 2048w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-1920x1048.png 1920w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-1170x639.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-01-585x319.png 585w\" sizes=\"(max-width: 2239px) 100vw, 2239px\" \/><\/a>\n<\/p>\n\n\n--><\/p>\n<p>\ud55c\ud3b8 (1)\uc758 \uadf9\ud55c\uc774 \ubc1c\uc0b0\ud558\uba74 \u201c\uad6c\uac04 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc774\uc0c1\uc801\ubd84\uc774 \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. (1)\uc758 \uadf9\ud55c\uc774 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0<br \/>\n\\[\\int_{a}^{\\infty} f(x) dx = \\infty\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b4\uba70, (1)\uc758 \uadf9\ud55c\uc774 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0<br \/>\n\\[\\int_{a}^{\\infty} f(x) dx = &#8211; \\infty\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. (1)\uc758 \uadf9\ud55c\uc774 \uc9c4\ub3d9\ud558\ub294 \uacbd\uc6b0\uc5d0\ub294 \uadf8\uac83\uc744 \uae30\ud638\ub85c \ub098\ud0c0\ub0b4\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\ud3b8\uc758\uc0c1 \u201c\\([a,\\,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4\u201d\ub77c\ub294 \ud45c\ud604\uc744 \u201c\uc774\uc0c1\uc801\ubd84 \\(\\int_a^\\infty f(x)dx\\)\uac00 \uc218\ub834\ud55c\ub2e4\u201d\ub77c\uace0 \ud45c\ud604\ud558\uae30\ub3c4 \ud558\uba70,  \u201c\\([a,\\,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \ubc1c\uc0b0\ud55c\ub2e4\u201d\ub77c\ub294 \ud45c\ud604\uc744 \u201c\uc774\uc0c1\uc801\ubd84 \\(\\int_a^\\infty f(x)dx\\)\uac00 \ubc1c\uc0b0\ud55c\ub2e4\u201d\ub77c\uace0 \ud45c\ud604\ud558\uae30\ub3c4 \ud55c\ub2e4. \uc989 (2)\uc640 \uac19\uc740 \ud45c\ud604\uc740 \uc774\uc0c1\uc801\ubd84 \uadf8 \uc790\uccb4\ub97c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud558\uace0 \uc218\ub834\ud558\ub294 \uc774\uc0c1\uc801\ubd84\uc758 \uac12\uc744 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_1 ^\\infty \\frac{1}{x^2} dx\\tag{3}\\]<br \/>\n\\(f(x)=\\frac{1}{x^2}\\)\uc774\ub77c\uace0 \ud558\uba74, \\(f\\)\ub294 \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{b\\rightarrow\\infty} \\int_1^b \\frac{1}{x^2} dx<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\left[ &#8211; \\frac{1}{x} \\right]_1 ^b \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\left( &#8211; \\frac{1}{b} +1 \\right) = 1<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c, \uc774\uc0c1\uc801\ubd84 (3)\uc740 \uc218\ub834\ud558\uace0 \uadf8 \uac12\uc740<br \/>\n\\[\\int_1 ^\\infty \\frac{1}{x^2} dx = 1\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><br \/>\n\ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_1 ^\\infty \\frac{1}{x} dx\\tag{4}\\]<br \/>\n\\(f(x)=\\frac{1}{x}\\)\uc774\ub77c\uace0 \ud558\uba74, \\(f\\)\ub294 \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{b\\rightarrow\\infty} \\int_1^b \\frac{1}{x} dx<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\bigg[ \\ln x \\bigg]_1 ^b \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\ln b = \\infty<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c, \uc774\uc0c1\uc801\ubd84 (4)\ub294 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p>\uc801\ubd84 \uad6c\uac04\uc774 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0\uc758 \uc774\uc0c1\uc801\ubd84\ub3c4 \uc55e\uc5d0\uc11c\uc640 \uac19\uc740 \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div>\n<a href=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02.png\" data-rel=\"penci-gallery-image-content\" ><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02.png\" alt=\"\" width=\"359\" height=\"204\" class=\"aligncenter size-full wp-image-8699\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02.png 2159w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-300x170.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-1024x580.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-768x435.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-1536x869.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-2048x1159.png 2048w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-1920x1087.png 1920w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-1170x663.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-585x331.png 585w\" sizes=\"(max-width: 359px) 100vw, 359px\" \/><\/a>\n<\/div>\n<p><!--\n\n\n<p>\n<a href=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02.png\" data-rel=\"penci-gallery-image-content\" ><img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02.png\" alt=\"\" width=\"2159\" height=\"1222\" class=\"aligncenter size-full wp-image-8699\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02.png 2159w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-300x170.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-1024x580.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-768x435.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-1536x869.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-2048x1159.png 2048w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-1920x1087.png 1920w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-1170x663.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-02-585x331.png 585w\" sizes=\"(max-width: 2159px) 100vw, 2159px\" \/><\/a>\n<\/p>\n\n\n--><\/p>\n<p>\ud568\uc218 \\(g\\)\uac00 \uad6c\uac04 \\((-\\infty ,\\,\\, b ]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc774\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud560 \ub54c, \uadf9\ud55c<br \/>\n\\[\\lim_{a\\rightarrow -\\infty} \\int_a^b g(x) dx\\tag{5}\\]<br \/>\n\ub85c \uc815\uc758\ub418\ub294 \uc774\uc0c1\uc801\ubd84\uc744<br \/>\n\\[\\int_{-\\infty}^b g(x) dx\\tag{6}\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\uc801\ubd84 \uad6c\uac04\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uace0 \uc544\ub798\ub85c\ub3c4 \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0\uc758 \uc774\uc0c1\uc801\ubd84\uc740 \ub450 \uc774\uc0c1\uc801\ubd84\uc758 \ud569\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc989 \ud568\uc218 \\(h\\)\uac00 \\((-\\infty ,\\,\\, \\infty )\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uc774\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud560 \ub54c, \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_{-\\infty} ^{\\infty} h(x) dx\\tag{7}\\]<br \/>\n\ub97c, \uc801\ub2f9\ud55c \uc810 \\(c\\)\ub97c \uae30\uc900\uc73c\ub85c \uad6c\uac04\uc744 \uc790\ub978 \ub450 \uc801\ubd84\uc758 \ud569<br \/>\n\\[\\int_{-\\infty} ^{c} h(x) dx + \\int_{c} ^{\\infty} h(x) dx\\tag{8}\\]<br \/>\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc5ec\uae30\uc11c (8)\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \ubaa8\ub450 \uc218\ub834\ud560 \ub54c\ub9cc (7)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud558\uace0, \uadf8 \uc774\uc678\uc5d0\ub294 (7)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \ubc1c\uc0b0\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\ud55c \uc810 \\(c\\)\uc5d0 \ub300\ud558\uc5ec (8)\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4\uba74, \\(c\\)\uc758 \uac12\uc744 \ub2e4\ub978 \uac83\uc73c\ub85c \ubc14\uafb8\uc5b4\ub3c4 (8)\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc740 \ubaa8\ub450 \uc218\ub834\ud558\uba70, \\(c\\)\uc758 \uac12\uacfc \uc0c1\uad00 \uc5c6\uc774 (8)\uc758 \uac12\uc740 \uc77c\uc815\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (8)\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294 \uacbd\uc6b0 \uc774\uc0c1\uc801\ubd84 (7)\uc758 \uac12\uc740 \\(c\\)\uc758 \uac12\uacfc \uc0c1\uad00 \uc5c6\uc774 \uc798 \uc815\uc758\ub41c\ub2e4.<\/p>\n<p>(7)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uac70\ub098 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0\ub97c \uc815\uc758\ud558\ub294 \uac83\uc740 \uc0c1\ub2f9\ud788 \uae4c\ub2e4\ub86d\ub2e4. \uc65c\ub0d0\ud558\uba74 (7)\uc740 \ub450 \uadf9\ud55c\uc758 \ud569\uc73c\ub85c \uc815\uc758\ub418\uae30 \ub54c\ubb38\uc774\ub2e4. \uc774 \uae00\uc5d0\uc11c\ub294 \uc815\uc758\ud558\uc9c0 \uc54a\uaca0\ub2e4. [\ub9cc\uc57d \uc774 \uae00\uc744 \ubcf4\uace0 (7)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0\uc640 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0\ub97c \uba85\ud655\ud558\uac8c \uc815\uc758\ud558\uace0 \uc2f6\ub2e4\ub294 \uc0dd\uac01\uc774 \ub4e4\uc5c8\ub2e4\uba74 \uc218\ud559 \uc804\uacf5\uc744 \ucd94\ucc9c\ud55c\ub2e4.]<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><br \/>\n\ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_{-\\infty}^{\\infty} \\frac{1}{1+x^2} dx\\tag{9}\\]<br \/>\n\\(h(x)=\\frac{1}{1+x^2}\\)\uc774\ub77c\uace0 \ud558\uba74, \\(h\\)\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \\(\\mathbb{R}\\)\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{a\\rightarrow -\\infty} \\int_a^0 \\frac{1}{1+x^2} dx<br \/>\n&#038;= \\lim_{a\\rightarrow -\\infty} \\bigg[ \\arctan x \\bigg]_a^0 \\\\[6pt]<br \/>\n&#038;= \\lim_{a\\rightarrow -\\infty} (-\\arctan a ) = \\frac{\\pi}{2}<br \/>\n\\end{align}\\]<br \/>\n\uc774\uace0<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{b\\rightarrow\\infty} \\int_0^b \\frac{1}{1+x^2} dx<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\bigg[ \\arctan x \\bigg]_0^b \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} (\\arctan b ) = \\frac{\\pi}{2}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\int_{-\\infty}^{\\infty} \\frac{1}{1+x^2} dx = \\frac{\\pi}{2} + \\frac{\\pi}{2} = \\pi\\]<br \/>\n\uc774\ub2e4. \uc989 \uc774\uc0c1\uc801\ubd84 (9)\ub294 \uc218\ub834\ud558\uace0, \uadf8 \uac12\uc740 \\(\\pi\\)\uc774\ub2e4. (\uc5ec\uae30\uc11c \\(\\arctan\\)\ub294 \ud0c4\uc820\ud2b8\uc758 \uc5ed\ud568\uc218\ub97c \ub098\ud0c0\ub0b8\ub2e4. \u2018\uc544\ud06c\ud0c4\uc820\ud2b8\u2019\ub77c\uace0 \uc77d\ub294\ub2e4.)\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><br \/>\n\ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_{-\\infty}^{\\infty} \\frac{2x}{1+x^2} dx\\tag{10}\\]<br \/>\n\\(h(x)=\\frac{2x}{1+x^2}\\)\uc774\ub77c\uace0 \ud558\uba74, \\(h\\)\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \\(\\mathbb{R}\\)\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uc774\uc81c (10)\uc758 \uc801\ubd84\uad6c\uac04\uc744 \\(0\\)\uc744 \uae30\uc900\uc73c\ub85c \ub450 \uad6c\uac04\uc73c\ub85c \uc798\ub77c\uc11c \uc774\uc0c1\uc801\ubd84\uc744 \uacc4\uc0b0\ud574 \ubcf4\uc790.<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{a\\rightarrow -\\infty} \\int_a^0 \\frac{2x}{1+x^2} dx<br \/>\n&#038;= \\lim_{a\\rightarrow -\\infty} \\bigg[ \\ln (1+x^2 ) \\bigg]_a^0 \\\\[6pt]<br \/>\n&#038;= \\lim_{a\\rightarrow -\\infty} (-\\ln (1+a^2 )) = -\\infty<br \/>\n\\end{align}\\]<br \/>\n\uc774\uace0<br \/>\n\\[\\begin{aligned}<br \/>\n\\lim_{b\\rightarrow \\infty} \\int_0^b \\frac{2x}{1+x^2} dx<br \/>\n&#038;= \\lim_{b\\rightarrow \\infty} \\bigg[ \\ln (1+x^2 ) \\bigg]_0^b \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow \\infty} (\\ln (1+b^2 )) = \\infty<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \uc774\uc0c1\uc801\ubd84 (10)\uc740 \ubc1c\uc0b0\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc704 \ubcf4\uae30 4\uc5d0\uc11c \uc8fc\uc758\ud560 \uc810\uc774 \uc788\ub2e4. \ud568\uc218<br \/>\n\\[h(x)=\\frac{2x}{1+x^2}\\]<br \/>\n\uac00 \uae30\ud568\uc218(\uadf8\ub798\ud504\uac00 \uc6d0\uc810\uc5d0 \ub300\ud558\uc5ec \ub300\uce6d\uc778 \ud568\uc218)\uc774\ubbc0\ub85c \uad6c\uac04 \\((-\\infty ,\\,\\, \\infty )\\)\uc5d0\uc11c \ud568\uc218 \\(h\\)\ub97c \uc801\ubd84\ud558\uba74 \uadf8 \uac12\uc774 \\(0\\)\uc774 \ub420 \uac83 \uac19\uc740 \u2018\ub290\ub08c\u2019\uc774 \ub4e0\ub2e4. \uc989<br \/>\n\\[\\begin{aligned}<br \/>\n\\lim_{b\\rightarrow\\infty} \\int_{-b}^{b} \\frac{2x}{1+x^2} dx<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\bigg[ \\ln (1+x^2 ) \\bigg]_{-b}^{b} \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} (\\ln (1+b^2 ) &#8211; \\ln (1+b^2 )) \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} 0 = 0<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\int_{-\\infty}^{\\infty} \\frac{2x}{1+x^2} dx = 0\\]<br \/>\n\uc77c \uac83 \uac19\ub2e4. \ud558\uc9c0\ub9cc \uc774\ub7ec\ud55c \uacc4\uc0b0\uc740 \uc633\uc9c0 \uc54a\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc774\uc911\uadf9\ud55c<br \/>\n\\[\\lim_{\\begin{gathered}a\\rightarrow -\\infty \\\\ b\\rightarrow\\infty \\end{gathered}} \\int_a^b h(x) dx \\]<br \/>\n\uc5d0\uc11c \\(a\\)\uac00 \uc791\uc544\uc9c0\ub294 \uc18d\ub3c4\uc640 \\(b\\)\uac00 \ucee4\uc9c0\ub294 \uc18d\ub3c4\uac00 \ud56d\uc0c1 \uc77c\uce58\ud558\ub294 \uac83\uc740 \uc544\ub2c8\uae30 \ub54c\ubb38\uc774\ub2e4. \uc774\ub7ec\ud55c \uc774\uc720 \ub54c\ubb38\uc5d0 (8)\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \ubaa8\ub450 \uc218\ub834\ud560 \ub54c\ub9cc (7)\uc774 \uc218\ub834\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"unboundedfunction\"><\/a><\/p>\n<h3>\uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\uc758 \uc774\uc0c1\uc801\ubd84<\/h3>\n<p>\\(a\\)\uc640 \\(b\\)\uac00 \uc2e4\uc218\uc774\uace0 \\(a < b\\)\uc774\uba70 \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\((a,\\,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \\(f\\)\uac00 \\((a,\\,\\,b]\\)\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\(f\\)\uac00 \\(a\\)\uc758 \uadfc\ucc98\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uadf9\ud55c\n\\[\\lim_{c\\rightarrow a+} \\int_{c}^{b} f(x) dx\\tag{11}\\]\n\uac00 \uc218\ub834\ud558\uba74, \u201c\uad6c\uac04 \\((a,\\,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4<\/span>\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \uadf8 \uadf9\ud55c\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\int_{a}^{b} f(x) dx \\tag{12}\\]<br \/>\n\uc774 \uadf9\ud55c\uc758 \uac12\uc744 \\((a,\\,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc774\uc0c1\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div>\n<a href=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03.png\" data-rel=\"penci-gallery-image-content\" ><img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03.png\" alt=\"\" width=\"288\" height=\"295\" class=\"aligncenter size-full wp-image-8700\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03.png 1728w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-293x300.png 293w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-1001x1024.png 1001w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-768x786.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-1501x1536.png 1501w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-1170x1197.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-585x599.png 585w\" sizes=\"(max-width: 288px) 100vw, 288px\" \/><\/a>\n<\/div>\n<p><!--\n\n\n<p>\n<a href=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03.png\" data-rel=\"penci-gallery-image-content\" ><img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03.png\" alt=\"\" width=\"1728\" height=\"1768\" class=\"aligncenter size-full wp-image-8700\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03.png 1728w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-293x300.png 293w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-1001x1024.png 1001w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-768x786.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-1501x1536.png 1501w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-1170x1197.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-03-585x599.png 585w\" sizes=\"(max-width: 1728px) 100vw, 1728px\" \/><\/a>\n<\/p>\n\n\n--><\/p>\n<p>(12)\uc640 \uac19\uc740 \uc801\ubd84\uc5d0\uc11c \uc810 \\(a\\)\ub97c \uc774\uc0c1\uc801\ubd84 (12)\uc758 <span class=\"defined\">\ud2b9\uc774\uc810<\/span>(singularity)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ud55c\ud3b8 (11)\uc758 \uadf9\ud55c\uc774 \ubc1c\uc0b0\ud558\uba74 \u201c\uad6c\uac04 \\((a,\\,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc774\uc0c1\uc801\ubd84\uc774 \ubc1c\uc0b0\ud55c\ub2e4<\/span>\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. (11)\uc758 \uadf9\ud55c\uc758 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0<br \/>\n\\[\\int_{a}^{b} f(x) dx = \\infty\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b4\uba70, (11)\uc758 \uadf9\ud55c\uc758 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0<br \/>\n\\[\\int_{a}^{b} f(x) dx = &#8211; \\infty\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ud3b8\uc758\uc0c1 \u201c\\((a,\\,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4\u201d\ub77c\ub294 \ud45c\ud604\uc744 \u201c\\([a,\\,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4\u201d\ub77c\uace0 \ud45c\ud604\ud558\uae30\ub3c4 \ud558\uace0 \u201c\uc774\uc0c1\uc801\ubd84 \\(\\int_{a}^{b} f(x) dx\\)\uac00 \uc218\ub834\ud55c\ub2e4\u201d\ub77c\uace0 \ud45c\ud604\ud558\uae30\ub3c4 \ud55c\ub2e4. \ubc1c\uc0b0\ud558\ub294 \uc774\uc0c1\uc801\ubd84\ub3c4 \uc774\uc640 \uac19\uc774 \uc735\ud1b5\uc131\uc788\uac8c \ud45c\ud604\ud55c\ub2e4. \uc989 (12)\uc640 \uac19\uc740 \ud45c\ud604\uc740 \uc774\uc0c1\uc801\ubd84 \uadf8 \uc790\uccb4\ub97c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud558\uace0 \uc218\ub834\ud558\ub294 \uc774\uc0c1\uc801\ubd84\uc758 \uac12\uc744 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.<\/span><br \/>\n\ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_0 ^1 \\frac{1}{\\sqrt{x}} dx\\tag{13}\\]<br \/>\n\\(f(x)=\\frac{1}{\\sqrt{x}}\\)\uc774\ub77c\uace0 \ud558\uba74, \ud568\uc218 \\(f\\)\ub294 \\(x=0\\)\uc5d0\uc11c \uc815\uc758\ub418\uc9c0 \uc54a\uc73c\uba70 \uad6c\uac04 \\((0,\\,\\,1]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4. \uc989 \\(x\\rightarrow 0+\\)\uc77c \ub54c \\(f(x) \\rightarrow \\infty\\)\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(f\\)\ub294 \\((0,\\,\\,1]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{c\\rightarrow 0+} \\int_c^1 \\frac{1}{\\sqrt{x}} dx<br \/>\n&#038;= \\lim_{c\\rightarrow 0+} \\bigg[ 2\\sqrt{x} \\bigg]_c ^1 \\\\[6pt]<br \/>\n&#038;= \\lim_{c\\rightarrow 0+} \\left( 2 &#8211; 2\\sqrt{c} \\right) = 2<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c, \uc774\uc0c1\uc801\ubd84 (13)\uc740 \uc218\ub834\ud558\uace0 \uadf8 \uac12\uc740 \\(2\\)\uc774\ub2e4. \uc989<br \/>\n\\[\\int_0 ^1 \\frac{1}{\\sqrt{x}} dx = 2\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.<\/span><br \/>\n\ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_0 ^1 \\frac{1}{x^2} dx\\tag{14}\\]<br \/>\n\\(f(x)=\\frac{1}{x^2}\\)\uc774\ub77c\uace0 \ud558\uba74, \ud568\uc218 \\(f\\)\ub294 \\(x=0\\)\uc5d0\uc11c \uc815\uc758\ub418\uc9c0 \uc54a\uc73c\uba70 \uad6c\uac04 \\((0,\\,\\,1]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4. \uc989 \\(x\\rightarrow 0+\\)\uc77c \ub54c \\(f(x) \\rightarrow \\infty\\)\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(f\\)\ub294 \\((0,\\,\\,1]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{c\\rightarrow 0+} \\int_c^1 \\frac{1}{x^2} dx<br \/>\n&#038;= \\lim_{c\\rightarrow 0+} \\left[ &#8211; \\frac{1}{x} \\right]_c ^1 \\\\[6pt]<br \/>\n&#038;= \\lim_{c\\rightarrow 0+} \\left( -1 + \\frac{1}{c} \\right) = \\infty<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c, \uc774\uc0c1\uc801\ubd84 (13)\uc740 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4. \uc989<br \/>\n\\[\\int_0 ^1 \\frac{1}{x^2} dx = \\infty\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc801\ubd84 \uad6c\uac04\uc758 \uc624\ub978\ucabd \ub05d\uc810\uc774 \ud2b9\uc774\uc810\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc55e\uc5d0\uc11c\uc640 \uac19\uc740 \ubc29\uc2dd\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div>\n<a href=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04.png\" data-rel=\"penci-gallery-image-content\" ><img loading=\"lazy\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04.png\" alt=\"\" width=\"288\" height=\"295\" class=\"aligncenter size-full wp-image-8701\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04.png 1728w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04-293x300.png 293w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04-1001x1024.png 1001w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04-768x786.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04-1501x1536.png 1501w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04-1170x1197.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2023\/05\/improper-integrals-04-585x599.png 585w\" sizes=\"(max-width: 288px) 100vw, 288px\" \/><\/a>\n<\/div>\n<p>\uc989 \ud568\uc218 \\(g\\)\uac00 \uad6c\uac04 \\([a,\\,\\,b)\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\uba70 \\(b\\) \uadfc\ucc98\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2d0 \ub54c, \uadf9\ud55c<br \/>\n\\[\\lim_{d\\rightarrow b-} \\int_a^d g(x) dx\\tag{15}\\]<br \/>\n\ub85c \uc815\uc758\ub418\ub294 \uc774\uc0c1\uc801\ubd84\uc744<br \/>\n\\[\\int_a^b g(x) dx\\tag{16}\\]<br \/>\n\uc640 \uae49\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ud2b9\uc774\uc810\uc774 \uc801\ubd84 \uad6c\uac04\uc758 \ub0b4\ubd80\uc5d0 \uc788\ub294 \uacbd\uc6b0\uc758 \uc774\uc0c1\uc801\ubd84\uc740 \ub450 \uc774\uc0c1\uc801\ubd84\uc758 \ud569\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc989 \\(a < u < b\\)\uc774\uace0 \ud568\uc218 \\(h\\)\uac00 \\([a ,\\,\\,u ) \\cup (u ,\\,\\, b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uc73c\uba70 \uc774 \uc9d1\ud569\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud560 \ub54c, \uc774\uc0c1\uc801\ubd84\n\\[\\int_{a}^{b} h(x) dx \\tag{17}\\]\n\ub97c \ub450 \uc801\ubd84\uc758 \ud569\n\\[\\int_{a}^{u} h(x) dx + \\int_{u}^{b} h(x) dx \\tag{18}\\]\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc5ec\uae30\uc11c (18)\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \ubaa8\ub450 \uc218\ub834\ud560 \ub54c\ub9cc (17)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud558\uace0, \uadf8 \uc678\uc5d0\ub294 (17)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \ubc1c\uc0b0\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 7.<\/span><br \/>\n\ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_0 ^2 \\frac{1}{\\sqrt{\\lvert x-1 \\rvert}} dx\\tag{19}\\]<br \/>\n\\(h(x) = \\lvert x-1 \\rvert ^{-1\/2}\\)\uc774\ub77c\uace0 \ud558\uba74, \ud568\uc218 \\(h\\)\ub294 \\(x=1\\)\uc5d0\uc11c \uc815\uc758\ub418\uc9c0 \uc54a\uc73c\uba70 \uc9d1\ud569 \\([0,\\,\\, 1)\\cup (1,\\,\\, 2]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4. \uc989 \\(x \\rightarrow 1-\\) \ub610\ub294 \\(x \\rightarrow 1+\\)\uc77c \ub54c \\(h(x) \\rightarrow \\infty\\)\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(h\\)\ub294 \\([0,\\,\\, 1)\\cup (1,\\,\\, 2]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uc9d1\ud569\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\uc774\uc81c \ubb38\uc81c\uc758 \uc801\ubd84\uc744 \ub450 \uc774\uc0c1\uc801\ubd84\uc758 \ud569\uc73c\ub85c \ub098\ud0c0\ub0b4\uc790.<br \/>\n\\[\\int_0 ^1 \\frac{1}{\\sqrt{\\lvert x-1 \\rvert}} dx + \\int_1 ^2 \\frac{1}{\\sqrt{\\lvert x-1 \\rvert}} dx \\tag{20}\\]<br \/>\n\\(x\\)\uc758 \ubc94\uc704\uc5d0 \ub530\ub77c \uc808\ub313\uac12\uc744 \ud480\uba74 \uc704 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\int_0 ^1 \\frac{1}{\\sqrt{1-x}} dx + \\int_1 ^2 \\frac{1}{\\sqrt{x-1}} dx \\tag{21}\\]<br \/>\n\uadf8\ub7f0\ub370<br \/>\n\\[\\begin{aligned}<br \/>\n\\int_0^1 \\frac{1}{\\sqrt{1-x}} dx<br \/>\n&#038;= \\lim_{d\\rightarrow 1-} \\int_0^d \\frac{1}{\\sqrt{1-x}} dx \\\\[4pt]<br \/>\n&#038;= \\lim_{d\\rightarrow 1-} \\bigg[ -2 \\sqrt{1-x} \\bigg]_0^d \\\\[4pt]<br \/>\n&#038;= \\lim_{d\\rightarrow 1-} \\left( -2 \\sqrt{1-d} +2 \\right) =2<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\uace0, \ub9c8\ucc2c\uac00\uc9c0\ub85c<br \/>\n\\[\\begin{aligned}<br \/>\n\\int_1^2 \\frac{1}{\\sqrt{x-1}} dx<br \/>\n&#038;= \\lim_{c\\rightarrow 1+} \\int_c^2 \\frac{1}{\\sqrt{x-1}} dx \\\\[4pt]<br \/>\n&#038;= \\lim_{c\\rightarrow 1+} \\bigg[ 2 \\sqrt{1-x} \\bigg]_c^2 \\\\[4pt]<br \/>\n&#038;= \\lim_{c\\rightarrow 1+} \\left( 2 \\sqrt{1-c} +2 \\right) =2<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4. \uc989 (21)\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \ubaa8\ub450 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (19)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uba70, \uadf8 \uac12\uc740<br \/>\n\\[\\int_0 ^2 \\frac{1}{\\sqrt{\\lvert x-1 \\rvert}} dx = 2+2 = 4\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"convergencetest1\"><\/a><\/p>\n<h3>\uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834 \ud310\uc815\ubc95 (\uc801\ubd84 \uad6c\uac04\uc758 \uae38\uc774\uac00 \ubb34\ud55c\uc778 \uacbd\uc6b0)<\/h3>\n<p>\uc774\uc0c1\uc801\ubd84\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\ub824\uba74 \ub2eb\ud78c \uad6c\uac04\uc5d0\uc11c \uc815\uc801\ubd84\uc744 \uad6c\ud55c \ub4a4 \uadf9\ud55c\uc744 \ucde8\ud574\uc57c \ud55c\ub2e4. \ud558\uc9c0\ub9cc \uc774\ub7ec\ud55c \ubc29\ubc95\uc73c\ub85c \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294\uc9c0 \uc5ec\ubd80\ub97c \ud310\uc815\ud558\ub294 \uac83\uc774 \ud56d\uc0c1 \uac00\ub2a5\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. \ub2e4\uc74c \uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_1^\\infty e^{-x^2} dx \\tag{22}\\]<br \/>\n\uc704 \uc801\ubd84\uc5d0\uc11c \ud53c\uc801\ubd84\ud568\uc218 \\(f(x) = e^{-x^2}\\)\uc758 \uc5ed\ub3c4\ud568\uc218\ub294 \ucd08\ub4f1\ud568\uc218(\uc720\ub9ac\ud568\uc218, \ubb34\ub9ac\ud568\uc218, \uc9c0\uc218\ud568\uc218, \ub85c\uadf8\ud568\uc218, \uc0bc\uac01\ud568\uc218\ub97c \uc720\ud55c \ubc88 \uacb0\ud569\ud558\uc5ec \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \ud568\uc218)\uac00 \uc544\ub2d8\uc774 \uc54c\ub824\uc838 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (22)\uc640 \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294\uc9c0 \ud310\uc815\ud558\ub824\uba74 \uc815\uc801\ubd84\uc744 \uc9c1\uc811 \uacc4\uc0b0\ud558\uc9c0 \uc54a\ub294 \ub2e4\ub978 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud574\uc57c \ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(F\\)\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\[F(x) = \\int_1 ^x e^{-t^2} dt \\quad (x \\ge 1 ) \\tag{23}\\]<br \/>\n\\(t \\ge 1\\)\uc77c \ub54c \\(e^{-t^2} > 0\\)\uc774\ubbc0\ub85c, \\(F\\)\ub294 \\([1 ,\\,\\,\\infty )\\)\uc5d0\uc11c \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ub2e4. \ud55c\ud3b8 \\(t\\ge 1\\)\uc77c \ub54c<br \/>\n\\[e^{-t^2} \\le e^{-t}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[\\lim_{x\\rightarrow\\infty} \\int_1^x e^{-t} dt = \\lim_{x\\rightarrow\\infty} \\left( \\frac{1}{e} &#8211; \\frac{1}{e^x}\\right) = \\frac{1}{e}\\]<br \/>\n\uc774\ubbc0\ub85c, \\(x \\ge 1\\)\uc778 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[F(x) = \\int_1^x e^{-t^2} dt \\le \\int_1^x e^{-t} dt \\le \\frac{1}{e}\\]<br \/>\n\uc774\ub2e4. \uc989 \\(F\\)\ub294 \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \uc720\uacc4\uc774\ub2e4. \uc774\ub85c\uc368 \ud568\uc218 \\(F\\)\uac00 \uad6c\uac04 \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \uc99d\uac00\ud558\uba74\uc11c \uc720\uacc4\uc784\uc744 \ubc1d\ud614\ub2e4. \uadf8\ub807\ub2e4\uba74 \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(F(x)\\)\uac00 \uc218\ub834\ud560\uae4c? \uadf8\uc5d0 \ub300\ud55c \ub2f5\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"lemma margintop2\">\n<p><span class=\"definition\">\ubcf4\uc870\uc815\ub9ac 1. (\ud568\uc218\uc758 \uadf9\ud55c\uc758 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac)<\/span><\/p>\n<p><p>\\(a\\)\uac00 \uc2e4\uc218\uc774\uace0 \ud568\uc218 \\(F\\)\uac00 \uad6c\uac04 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(F\\)\uac00 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc99d\uac00\ud558\uace0 \uc704\ub85c \uc720\uacc4\uc774\uba74, \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(F(x)\\)\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\ud568\uc218 \\(F\\)\uc5d0 \uc758\ud55c \\([a,\\,\\,\\infty )\\)\uc758 \uc0c1(image)\uc744 \\(E\\)\ub77c\uace0 \ub450\uc790. \uc989<br \/>\n\\[E = \\left\\{ F(x) \\,\\vert\\, x\\in [a,\\,\\,\\infty )\\right\\}\\]<br \/>\n\ub77c\uace0 \ub450\uc790. \uadf8\ub7ec\uba74 \\(E\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c, \uc2e4\uc218\uacc4\uc758 \uc644\ube44\uc131 \uacf5\ub9ac(\ucd5c\uc18c\uc0c1\uacc4 \uc131\uc9c8)\uc5d0 \uc758\ud558\uc5ec \\(E\\)\uc758 \uc0c1\ud55c(\ucd5c\uc18c\uc0c1\uacc4)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(E\\)\uc758 \uc0c1\ud55c\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\uc774\uc81c \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(F\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790. \uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[L &#8211; \\epsilon < F(X) \\le L\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(X\\)\uac00 \\([a,\\,\\,\\infty )\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc774\uc640 \uac19\uc740 \\(X\\)\uc5d0 \ub300\ud558\uc5ec, \\(x > X\\)\uc77c \ub54c<br \/>\n\\[L &#8211; \\epsilon < F(X) \\le F(x) \\le L\\]\n\uc774\ubbc0\ub85c\n\\[\\lvert F(x) - L \\rvert < \\epsilon\\]\n\uc774\ub2e4. \uc989 \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(X\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.\n\\[x > X \\quad \\Longrightarrow \\quad \\lvert F(x) &#8211; L \\rvert < \\epsilon .\\]\n\uadf8\ub7ec\ubbc0\ub85c \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(F(x)\\)\uac00 \\(L\\)\uc5d0 \uc218\ub834\ud55c\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>(23)\uc5d0\uc11c \uc815\uc758\ud55c \ud568\uc218 \\(F\\)\uac00 \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \uc99d\uac00\ud558\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c, \ubcf4\uc870\uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(F(x)\\)\ub294 \uc218\ub834\ud55c\ub2e4. \uc989<br \/>\n\\[\\lim_{x\\rightarrow\\infty} \\int_1^x e^{-t^2} dt\\]<br \/>\n\uac00 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_1^\\infty e^{-x^2} dx\\tag{22}\\]<br \/>\n\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc774\uc0c1\uc801\ubd84 (22)\uac00 \uc218\ub834\ud568\uc744 \uc99d\uba85\ud55c \uacfc\uc815\uc744 \uc77c\ubc18\ud654\ud558\uba74 \ub2e4\uc74c \ub450 \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\uc774\uc0c1\uc801\ubd84\uc758 \uc720\uacc4 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\([a,\\,\\,\\infty )\\)\uc758 \uc784\uc758\uc758 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\ge 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c<br \/>\n\\[F(x) = \\int_a ^x f(t) dt\\tag{24}\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^\\infty f(x) dx\\tag{25}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ud568\uc218 \\(F\\)\uac00 \\([a,\\,\\,\\infty)\\)\uc5d0\uc11c \uc720\uacc4\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\ud568\uc218 \\(F\\)\uac00 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \uac00\uc815\ud558\uc790. \\( x_1 < x_2 \\)\uc774\uace0 \\([a,\\,\\,\\infty )\\)\uc5d0 \uc18d\ud558\ub294 \ub450 \uc810 \\(x_1 ,\\) \\(x_2 \\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\begin{aligned}\nF(x_2 ) - F(x_1 ) &#038;= \\int_a^{x_2} f(t)dt - \\int_a^{x_1} f(t)dt \\\\[6pt]\n&#038;= \\int_{x_1}^{x_2} f(t)dt \\ge \\int_{x_1}^{x_2} 0\\, dt = 0\n\\end{aligned}\\]\n\uc989\n\\[F(x_1 ) \\le F(x_2 )\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(F\\)\ub294 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ub2e4. \ub530\ub77c\uc11c \ud568\uc218\uc758 \uadf9\ud55c\uc758 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac(\ubcf4\uc870\uc815\ub9ac 1)\uc5d0 \uc758\ud558\uc5ec, \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(F\\)\uac00 \uc218\ub834\ud55c\ub2e4.\n\\[\\int_a^\\infty f(x) dx = \\lim_{x\\rightarrow\\infty} F(x)\\]\n\uc774\ubbc0\ub85c \uc774\uc0c1\uc801\ubd84 (25)\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \ud568\uc218 \\(F\\)\uac00 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uc790. \\(F\\)\uac00 \uc99d\uac00\ud568\uc218\uc774\ubbc0\ub85c, \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(F(x)\\)\uac00 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud55c\ub2e4. \uc774\uac83\uc744 \ub354 \uc5c4\ubc00\ud558\uac8c \uc99d\uba85\ud574 \ubcf4\uc790.<\/p>\n<p>\uc591\uc218 \\(M\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \ud568\uc218 \\(F\\)\uac00 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ubbc0\ub85c \\(F(X) > M\\)\uc778 \uc2e4\uc218 \\(X\\)\uac00 \\([a,\\,\\,\\infty )\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \\(F\\)\uac00 \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ubbc0\ub85c<br \/>\n\\[x > X \\quad\\Longrightarrow\\quad F(x) > M\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(F(x) \\rightarrow\\infty\\)\uc774\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 \ud568\uc218 \\(F\\)\uac00 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\uba74 \uc774\uc0c1\uc801\ubd84 (25)\uac00 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4\ub294 \uc0ac\uc2e4\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\uc774\uc0c1\uc801\ubd84\uc758 \ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \uad6c\uac04 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\([a,\\,\\,\\infty )\\)\uc758 \uc784\uc758\uc758 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(0 \\le f(x) \\le g(x)\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^\\infty g(x) dx\\tag{26}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uba74, \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^\\infty f(x) dx\\tag{27}\\]<br \/>\n\ub3c4 \uc218\ub834\ud55c\ub2e4. [\ub2ec\ub9ac \ub9d0\ud558\uba74, \uac19\uc740 \uc870\uac74 \uc544\ub798\uc5d0\uc11c (27)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \ubc1c\uc0b0\ud558\uba74 (26)\uc758 \uc774\uc0c1\uc801\ubd84\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4.]<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc774\uc0c1\uc801\ubd84 (26)\uc774 \uc218\ub834\ud558\ubbc0\ub85c, (26)\uc758 \uc801\ubd84\uac12\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c \ud568\uc218 \\(F\\)\uc640 \\(G\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[\\begin{gathered}<br \/>\nF(x) = \\int_a^x f(t) dt \\quad (x\\in [a,\\,\\,\\infty )) , \\\\[6pt]<br \/>\nG(x) = \\int_a^x g(t) dt \\quad (x\\in [a,\\,\\,\\infty )) .<br \/>\n\\end{gathered}\\]<br \/>\n\uadf8\ub7ec\uba74 \\([a,\\,\\,\\infty )\\)\uc5d0 \uc18d\ud558\ub294 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[F(x) = \\int_a^x f(t) dt \\le \\int_a^x g(t) dt = G(x)\\]<br \/>\n\uc774\uace0, \\(G\\)\uac00 \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\uba70, \\(x\\rightarrow\\infty\\)\uc77c \ub54c \\(G(x)\\rightarrow L\\)\uc774\ubbc0\ub85c,<br \/>\n\\[F(a) \\le F(x) \\le G(x) \\le L\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c \ud568\uc218 \\(F\\)\ub294 \uad6c\uac04 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc720\uacc4\uc774\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \uc774\uc0c1\uc801\ubd84\uc758 \uc720\uacc4 \ud310\uc815\ubc95(\uc815\ub9ac 2)\uc5d0 \uc758\ud558\uc5ec \uc774\uc0c1\uc801\ubd84 (27)\uc774 \uc218\ub834\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc774\ubc88\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc0c1\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\int_{1}^{\\infty} \\frac{x+1}{x^3 + \\sin x^2} dx \\tag{28}\\]<br \/>\n\ud53c\uc801\ubd84\ud568\uc218<br \/>\n\\[f(x) = \\frac{x+1}{x^3 + \\sin x^2}\\tag{29}\\]<br \/>\n\uac00 \uad6c\uac04 \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c, \\(f\\)\ub294 \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7ec\ub098 \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub294 \uc77c\uc740 \uc27d\uc9c0 \uc54a\uc544 \ubcf4\uc778\ub2e4. \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub294 \ub300\uc2e0 \ube44\uad50 \ud310\uc815\ubc95(\uc815\ub9ac 3)\uc744 \uc0ac\uc6a9\ud558\uae30 \uc704\ud574 \\(f(x) \\le g(x)\\)\uc774\uba74\uc11c \\(\\int_1^\\infty g(x) dx\\)\uac00 \uc218\ub834\ud558\ub294 \ud568\uc218 \\(g\\)\ub97c \ucc3e\uac70\ub098, \\(0\\le h(x) \\le f(x)\\)\uc774\uba74\uc11c \\(\\int_1^\\infty h(x) dx\\)\uac00 \ubc1c\uc0b0\ud558\ub294 \ud568\uc218 \\(h\\)\ub97c \ucc3e\ub294 \uac83\uc744 \uc2dc\ub3c4\ud574\ubcfc \uc218 \uc788\ub2e4.<\/p>\n<p>\uc5ec\uae30\uc11c\ub294 \ub2e4\ub978 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud558\uaca0\ub2e4. (29)\uc758 \uc6b0\ubcc0\uc744 \ubcf4\uba74, \\(x\\rightarrow\\infty\\)\uc77c \ub54c \ubd84\uc790\ub294 \\(x\\)\uc640 \ube44\uc2b7\ud55c \uc815\ub3c4\ub85c \ucee4\uc9c0\uace0 \ubd84\ubaa8\ub294 \\(x^3\\)\uacfc \ube44\uc2b7\ud55c \uc815\ub3c4\ub85c \ucee4\uc9c4\ub2e4. \uc5ec\uae30\uc5d0\uc11c \ud78c\ud2b8\ub97c \uc5bb\uc5b4<br \/>\n\\[g(x) = \\frac{x}{x^3} = \\frac{1}{x^2}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x\\rightarrow\\infty\\)\uc77c \ub54c<br \/>\n\\[\\frac{f(x)}{g(x)} = \\frac{ (x+1) x^2 }{ x^3 + \\sin x^2  } \\,\\rightarrow \\, 1\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\)\uac00 \ucda9\ubd84\ud788 \ud070 \uac12\uc774\ub77c\uba74<br \/>\n\\[1 &#8211; \\frac{1}{2} < \\frac{f(x)}{g(x)} < 1 + \\frac{1}{2}\\]\n\uc774\ub77c\uace0 \ud560 \uc218 \uc788\ub2e4. \ub354 \uc815\ud655\ud558\uac8c \ub9d0\ud558\uba74, \\(\\epsilon = \\frac{1}{2}\\)\ub85c \ub450\uc5c8\uc744 \ub54c, \uc2e4\uc218 \\(X\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[x > X \\quad\\Longrightarrow\\quad \\left\\lvert \\frac{f(x)}{g(x)} \\right\\rvert < \\epsilon\\]\n\uc989\n\\[x > X \\quad\\Longrightarrow\\quad  1- \\frac{1}{2} < \\frac{f(x)}{g(x)} < 1+ \\frac{1}{2}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. (\uc5ec\uae30\uc11c \\(X\\)\ub294 \uad6c\uac04 \\([1,\\,\\,\\infty)\\)\uc758 \uc810\uc774\ub2e4.) \\(x > X\\)\uc77c \ub54c<br \/>\n\\[ f(x) < \\frac{3}{2} g(x)\\]\n\uc774\uace0, \uc774\uc0c1\uc801\ubd84\n\\[\\int_{X}^\\infty \\frac{3}{2} g(x) dx = \\int_{X}^\\infty \\frac{3}{2x^2} dx\\]\n\uac00 \uc218\ub834\ud558\ubbc0\ub85c, \uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec \uc774\uc0c1\uc801\ubd84\n\\[\\int_X^\\infty f(x) dx = \\int_{X}^{\\infty} \\frac{x+1}{x^3 + \\sin x^2} dx \\tag{29}\\]\n\ub3c4 \uc218\ub834\ud55c\ub2e4. \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([1,\\,\\,X]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c, \uc704 \uc774\uc0c1\uc801\ubd84\uc758 \uc544\ub798\ub05d\uc744 \\(X\\) \ub300\uc2e0 \\(1\\)\ub85c \ubc14\uafb8\uc5b4\ub3c4 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc740 \ubcc0\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc989\n\\[\\int_1^\\infty f(x)dx = \\int_1 ^X f(x) dx + \\int_X^\\infty f(x) dx\\]\n\uc774\uba70, \uc774 \uc774\uc0c1\uc801\ubd84\uc740 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ub0b4\uc6a9\uc744 \uc77c\ubc18\ud654\ud558\uba74 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 4. (\uc774\uc0c1\uc801\ubd84\uc758 \uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \uad6c\uac04 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\uba70, \\([a,\\,\\,\\infty)\\)\uc758 \uc784\uc758\uc758 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) > 0,\\) \\(g(x) > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \uadf9\ud55c<br \/>\n\\[\\lim_{x\\rightarrow\\infty} \\frac{f(x)}{g(x)}\\tag{30}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uace0 \uadf8 \uac12\uc774 \uc591\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^\\infty f(x) dx\\tag{31}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a ^\\infty g(x) dx\\tag{32}\\]<br \/>\n\uac00 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4. \uc989 \ub450 \uc774\uc0c1\uc801\ubd84 (30)\uacfc (31)\uc758 \uc218\ub834, \ubc1c\uc0b0 \uc5ec\ubd80\uac00 \uc77c\uce58\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uadf9\ud55c (30)\uc758 \uac12\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0<br \/>\n\\[\\epsilon = \\frac{L}{2}\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \\(L\\)\uc774 \uc591\uc218\uc774\ubbc0\ub85c \\(\\epsilon\\)\ub3c4 \uc591\uc218\uc774\ub2e4. \ud568\uc218\uc758 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec, \\([a,\\,\\,\\infty )\\)\uc5d0 \uc18d\ud558\ub294 \uc810 \\(X\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.<br \/>\n\\[x > X \\quad\\Longrightarrow\\quad \\left\\lvert \\frac{f(x)}{g(x)} &#8211; L \\right\\rvert < \\epsilon .\\]\n\uc989 \\(x > X\\)\uc77c \ub54c<br \/>\n\\[L &#8211; \\epsilon < \\frac{f(x)}{g(x)} < L+\\epsilon\\]\n\uc774\uba70, \uc774 \ubd80\ub4f1\uc2dd\uc744 \ubcc0\ud615\ud558\uba74\n\\[\\frac{L}{2} g(x) < f(x) < \\frac{3L}{2} g(x)\\tag{33}\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c, \ub9cc\uc57d \uad6c\uac04 \\([X,\\,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uba74 \uc815\ub9ac 3\uacfc (33)\uc758 \uccab \ubc88\uc9f8 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\ub3c4 \uc218\ub834\ud558\uba70, \ub9cc\uc57d \uad6c\uac04 \\([X,\\,\\,\\infty)\\)\uc5d0\uc11c \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uba74 \uc815\ub9ac 3\uacfc (33)\uc758 \ub450 \ubc88\uc9f8 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud558\uc5ec \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \uac01\uac01 \uad6c\uac04 \\([a,\\,\\,X]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c, \uad6c\uac04 \\([a,\\,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uad6c\uac04 \\([X,\\,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\uba70, \uad6c\uac04 \\([a,\\,\\,\\infty)\\)\uc5d0\uc11c \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uad6c\uac04 \\([X,\\,\\,\\infty)\\)\uc5d0\uc11c \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uc99d\uba85\ud55c \ub0b4\uc6a9\uc744 \uacb0\ud569\ud558\uba74 \uc815\ub9ac\uc758 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834, \ubc1c\uc0b0\uc744 \ud310\uc815\ud558\ub294 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 8.<\/span><br \/>\n\ub2e4\uc74c \uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834, \ubc1c\uc0b0\uc744 \ud310\uc815\ud574 \ubcf4\uc790.<br \/>\n\\[\\int_1^\\infty \\frac{x+1}{\\sqrt{x^5 +3} -1} dx\\tag{34}\\]<br \/>\n\\(x \\ge 1\\)\uc778 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 \\(f\\)\uc640 \\(g\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\nf(x) &#038;= \\frac{x+1}{\\sqrt{x^5 + 3}-1} , \\\\[6pt]<br \/>\ng(x) &#038;= x^{-\\frac{3}{2}} .<br \/>\n\\end{aligned}\\]<br \/>\n\uadf8\ub7ec\uba74 \\(x\\ge 1\\)\uc77c \ub54c \\(f(x) > 0,\\) \\(g(x) > 0\\)\uc774\uace0,<br \/>\n\\(x\\rightarrow\\infty\\)\uc77c \ub54c<br \/>\n\\[\\begin{aligned}<br \/>\n\\frac{f(x)}{g(x)} &#038;= \\frac{ x^{\\frac{5}{2}} + x^{\\frac{3}{2}} }{ \\sqrt{x^5 + 3}-1 }<br \/>\n= \\frac{ 1+x^{-1} }{ \\sqrt{ 1 + 3x^{-5} } &#8211; x^{ -\\frac{5}{2} } } \\,\\rightarrow\\,1<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[\\begin{aligned}<br \/>\n\\int_1^\\infty g(x) dx<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\int_1^b x^{-\\frac{3}{2}} \\,dx \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\left[ -\\frac{2}{\\sqrt{x}} \\right]_1^b \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\left( -\\frac{2}{\\sqrt{b}} +2 \\right) = 2<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c \\([1,\\,\\,\\infty )\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4. \uc989 \uc774\uc0c1\uc801\ubd84 (34)\uac00 \uc218\ub834\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 9.<\/span><br \/>\n\ub2e4\uc74c \uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834, \ubc1c\uc0b0\uc744 \ud310\uc815\ud574 \ubcf4\uc790.<br \/>\n\\[\\int_3^\\infty \\frac{1}{2x+\\sin x} dx\\tag{35}\\]<br \/>\n\\(x\\ge 3\\)\uc778 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 \\(f\\)\uc640 \\(g\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nf(x) &#038;= \\frac{1}{2x+\\sin x} , \\\\[6pt]<br \/>\ng(x) &#038;= \\frac{1}{x} .<br \/>\n\\end{aligned}\\]<br \/>\n\uadf8\ub7ec\uba74 \\(x\\ge 3\\)\uc77c \ub54c \\(f(x) > 0, \\) \\(g(x) > 0\\)\uc774\uace0, \\(x\\rightarrow\\infty\\)\uc77c \ub54c<br \/>\n\\[\\frac{f(x)}{g(x)} = \\frac{x}{2x+\\sin x} \\,\\rightarrow\\, \\frac{1}{2}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\([3,\\,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\([3,\\,\\,\\infty)\\)\uc5d0\uc11c \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[\\int_3^\\infty g(x) dx = \\int_3^\\infty \\frac{1}{x} dx = \\infty\\]<br \/>\n\uc774\ubbc0\ub85c, \\([3,\\,\\,\\infty )\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\ub3c4 \ubc1c\uc0b0\ud55c\ub2e4. \uc989 \uc774\uc0c1\uc801\ubd84 (35)\uac00 \ubc1c\uc0b0\ud55c\ub2e4.\n<\/p>\n<\/div>\n<p>\uc815\ub9ac 4\uc758 \uc99d\uba85 \uacfc\uc815\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \ub530\ub984\uc815\ub9ac\ub97c \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"corollary\">\n<p><span class=\"corollary\">\ub530\ub984\uc815\ub9ac 5. (\uc774\uc0c1\uc801\ubd84\uc758 \uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n<ol class=\"bracket marginbottom0\">\n<li>\uc815\ub9ac 4\uc5d0\uc11c \uadf9\ud55c (30)\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uace0 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uba74, \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\uc815\ub9ac 4\uc5d0\uc11c \uadf9\ud55c (30)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uace0 \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uba74, \\([a,\\,\\,\\infty )\\)\uc5d0\uc11c \\(g\\)\uc758 \uc774\uc0c1\uc801\ubd84\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"convergencetest2\"><\/a><\/p>\n<h3>\uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834 \ud310\uc815\ubc95 (\ud568\uc218\uac00 \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0)<\/h3>\n<p>\ubcf4\uc870\uc815\ub9ac 1\uacfc \uc815\ub9ac 2~4\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\uc758 \uc774\uc0c1\uc801\ubd84\uc5d0 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub294 \ud615\ud0dc\ub85c \ubc14\uafc0 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"lemma\">\n<p><span class=\"lemma\">\ubcf4\uc870\uc815\ub9ac 6. (\ud568\uc218\uc758 \uadf9\ud55c\uc758 \ub2e8\uc870\uc218\ub834 \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(F\\)\uac00 \uc5f4\ub9ac\uad6c\uac04 \\((a,\\,\\,b)\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uc99d\uac00\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket marginbottom0\">\n<li>\ub9cc\uc57d \\(F\\)\uac00 \\((a,\\,\\,b)\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uc774\uba74 \\(x \\rightarrow b -\\)\uc77c \ub54c \\(F(x)\\)\uac00 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(F\\)\uac00 \\((a,\\,\\,b)\\)\uc5d0\uc11c \uc544\ub798\ub85c \uc720\uacc4\uc774\uba74 \\(x \\rightarrow a +\\)\uc77c \ub54c \\(F(x)\\)\uac00 \uc218\ub834\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc5ec\uae30\uc11c\ub294 [1]\ub9cc \uc99d\uba85\ud55c\ub2e4.<br \/>\n\\[E = \\left\\{ F(x) \\,\\vert\\, x\\in (a,\\,\\,b)\\right\\}\\]<br \/>\n\ub77c\uace0 \ub450\uc790. \uadf8\ub7ec\uba74 \\(E\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc704\ub85c \uc720\uacc4\uc774\ubbc0\ub85c, \uc2e4\uc218\uacc4\uc758 \uc644\ube44\uc131 \uacf5\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(E\\)\uc758 \uc0c1\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(E\\)\uc758 \uc0c1\ud55c\uc744 \\(L\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[L- \\epsilon < F(X) \\le L\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(X\\)\uac00 \uad6c\uac04 \\((a,\\,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \\(\\delta = b-X\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\delta\\)\ub294 \uc591\uc218\uc774\ub2e4. \ub610\ud55c\n\\[b-\\delta < x < b \\quad\\Longrightarrow\\quad X < x < b \\quad\\Longrightarrow\\quad \\lvert F(x)-L \\rvert < \\epsilon\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x\\rightarrow b-\\)\uc77c \ub54c \\(F(x) \\rightarrow L\\)\uc774\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub2e4\uc74c \uc138 \uc815\ub9ac\ub294 \uc99d\uba85\ud558\uc9c0 \uc54a\uace0 \uc18c\uac1c\ud55c\ub2e4. (\uadf8\ub807\ub2e4\uace0 \ud574\uc11c \uc99d\uba85\uc744 \ud558\uc9c0 \uc54a\uace0 \uc0ac\uc6a9\ud574\ub3c4 \ub41c\ub2e4\ub294 \ub73b\uc740 \uc544\ub2c8\ubbc0\ub85c, \uc9c1\uc811 \uc99d\uba85\ud574 \ubcf4\uae30 \ubc14\ub780\ub2e4.) \ub2e4\uc74c \uc138 \uc815\ub9ac\uc5d0\uc11c \\(a < b\\)\uc778 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 7. (\uc774\uc0c1\uc801\ubd84\uc758 \uc720\uacc4 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,\\,b)\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\([a,\\,\\,b)\\)\uc758 \uc784\uc758\uc758 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\ge 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c<br \/>\n\\[F(x) = \\int_a^x f(t) dt \\tag{36}\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^b f(x) dx \\tag{37}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \ud568\uc218 \\(F\\)\uac00 \\([a,\\,\\,b)\\)\uc5d0\uc11c \uc704\ub85c \uc720\uacc4\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc815\ub9ac 2\uc758 \uc99d\uba85\uacfc \ube44\uc2b7\ud558\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 8. (\uc774\uc0c1\uc801\ubd84\uc758 \ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \uad6c\uac04 \\([a,\\,\\,b)\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\([a,\\,\\,b)\\)\uc758 \uc784\uc758\uc758 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(0\\le f(x) \\le g(x)\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^b g(x) dx \\tag{38}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uba74, \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^b f(x) dx \\tag{39}\\]<br \/>\n\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc815\ub9ac 3\uc758 \uc99d\uba85\uacfc \ube44\uc2b7\ud558\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 9. (\uc774\uc0c1\uc801\ubd84\uc758 \uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \uad6c\uac04 \\([a,\\,\\,b)\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\uba70, \\([a,\\,\\,b)\\)\uc758 \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) > 0,\\) \\(g(x) > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \uadf9\ud55c<br \/>\n\\[\\lim_{x\\rightarrow b} \\frac{f(x)}{g(x)} \\tag{40}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uace0 \uadf8 \uac12\uc774 \uc591\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^b f(x) dx \\tag{41}\\]<br \/>\n\uac00 \uc218\ub834\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_a^b g(x) dx \\tag{42}\\]<br \/>\n\uac00 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc815\ub9ac 4\uc758 \uc99d\uba85\uacfc \ube44\uc2b7\ud558\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"corollary\">\n<p><span class=\"corollary\">\ub530\ub984\uc815\ub9ac 10. (\uc774\uc0c1\uc801\ubd84\uc758 \uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95)<\/span><\/p>\n<ol class=\"bracket marginbottom0\">\n<li>\uc815\ub9ac 9\uc5d0\uc11c \uadf9\ud55c (40)\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud558\uace0 \uc774\uc0c1\uc801\ubd84 (42)\uac00 \uc218\ub834\ud558\uba74 \uc774\uc0c1\uc801\ubd84 (41)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/li>\n<li>\uc815\ub9ac 4\uc5d0\uc11c \uadf9\ud55c (40)\uc774 \uc591\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\uace0 \uc774\uc0c1\uc801\ubd84 (41)\uc774 \uc218\ub834\ud558\uba74 \uc774\uc0c1\uc801\ubd84 (42)\ub3c4 \uc218\ub834\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 10.<\/span><br \/>\n\ub2e4\uc74c \uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834, \ubc1c\uc0b0\uc744 \ud310\uc815\ud574 \ubcf4\uc790.<br \/>\n\\[\\int_0^{2} \\frac{1}{\\sqrt{\\sin x}} dx\\tag{43}\\]<br \/>\n\\(0 < x \\le 2\\)\uc778 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 \\(f\\)\uc640 \\(g\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.\n\\[\\begin{aligned}\nf(x) &#038;= \\frac{1}{\\sqrt{\\sin x}} ,\\\\[6pt]\ng(x) &#038;= \\frac{1}{\\sqrt{x}} .\n\\end{aligned}\\]\n\uadf8\ub7ec\uba74 \ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\ub294 \\((0 ,\\,\\, 2]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uad6c\uac04\uc5d0\uc11c \uad6d\uc18c\uc801\uc73c\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c \\(0 < x \\le 2\\)\uc77c \ub54c \\(f(x) > 0,\\) \\(g(x) > 0\\)\uc774\uace0, \\(x\\rightarrow 0+\\)\uc77c \ub54c<br \/>\n\\[\\frac{f(x)}{g(x)} = \\sqrt{ \\frac{x}{\\sin x} } \\,\\rightarrow\\, 1\\]<br \/>\n\uc774\ub2e4. \ud55c\ud3b8 \uc774\uc0c1\uc801\ubd84<br \/>\n\\[\\int_0^2 g(x) dx = \\int_0^2 \\frac{1}{\\sqrt{x}}dx\\]<br \/>\n\uac00 \uc218\ub834\ud558\ubbc0\ub85c, \uadf9\ud55c\ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\((0,\\,\\,2]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\ub3c4 \uc218\ub834\ud55c\ub2e4. \uc989 \uc774\uc0c1\uc801\ubd84 (43)\uc774 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"gammafunction\"><\/a><\/p>\n<h3>\uc774\uc0c1\uc801\ubd84\uc744 \ud65c\uc6a9\ud558\ub294 \uc608: \uac10\ub9c8 \ud568\uc218<\/h3>\n<p>\uc774\uc0c1\uc801\ubd84\uc744 \ud65c\uc6a9\ud558\ub294 \uc608\ub85c\uc11c \uac10\ub9c8 \ud568\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uac10\ub9c8 \ud568\uc218\ub294 \uc591\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(\\Gamma\\)\uc774\ub2e4.<br \/>\n\\[\\Gamma (x) = \\int_0^\\infty t^{x-1} e^{-t} dt .\\tag{44}\\]<br \/>\n\uac10\ub9c8 \ud568\uc218\ub294 \ucc28\ub840\uacf1 \\(n!\\)\uc744 \uc2e4\uc218 \ubc94\uc704\ub85c \ud655\uc7a5\ud55c \ud568\uc218\uc774\ub2e4. (\ub2e8, \ubaa8\ub4e0 \uc2e4\uc218\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ub418\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4.) \uc6b0\ub9ac\uc758 \ubaa9\ud45c\ub294 \uc784\uc758\uc758 \uc591\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uac10\ub9c8 \ud568\uc218\uac00 \uc218\ub834\ud55c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \ubc1d\ud788\uace0, \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(\\Gamma (n) = (n-1)!\\)\uc784\uc744 \ubc1d\ud788\ub294 \uac83\uc774\ub2e4.<\/p>\n<p>\ubcc4\ub2e4\ub978 \uc5b8\uae09\uc774 \uc5c6\uc73c\uba74 \uac10\ub9c8 \ud568\uc218\uc758 \uc124\uba85\uc774 \ub05d\ub0a0 \ub54c\uae4c\uc9c0 \\(x\\)\ub294 \uc591\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n<p>\uc6b0\uc120 \ub2e4\uc74c\uacfc \uac19\uc740 \uadf9\ud55c\uc744 \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\lim_{t\\rightarrow\\infty} \\frac{t^{x-1}}{e^{t\/2}}\\tag{45}\\]<br \/>\n\\(t\\rightarrow\\infty\\)\uc77c \ub54c, \ubc11\uc774 \\(1\\)\ubcf4\ub2e4 \ud070 \uc9c0\uc218\ud568\uc218\ub294 \uc5b4\ub5a0\ud55c \ub2e4\ud56d\ud568\uc218\ubcf4\ub2e4\ub3c4 \ube60\ub974\uac8c \uc99d\uac00\ud558\ubbc0\ub85c, \uc704 \uadf9\ud55c (45)\ub294 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74\uc11c \\(1\\)\ubcf4\ub2e4 \ud070 \uc2e4\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[t \\ge N \\quad\\Longrightarrow\\quad \\frac{t^{x-1}}{e^{t\/2}} \\le 1.\\]<br \/>\n\uc704 \uc2dd\uc758 \uc624\ub978\ucabd \ubd80\ub4f1\uc2dd\uc758 \uc591\ubcc0\uc5d0 \\(e^{-t\/2}\\)\uc744 \uacf1\ud558\uba74 \ub2e4\uc74c \uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[t \\ge N \\quad\\Longrightarrow\\quad t^{x-1} e^{-t} \\le e^{-t\/2}. \\tag{46}\\]<br \/>\n\uc774\uc81c \\(N\\)\uc744 \uace0\uc815\uc2dc\ud0a4\uace0, \\(\\Gamma\\)\ub97c \uc138 \uac1c\uc758 \uc801\ubd84\uc73c\ub85c \ucabc\uac1c\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\n\\Gamma_1 (x) &#038;= \\int_0^1 t^{x-1} e^{-t} dt , \\\\[6pt]<br \/>\n\\Gamma_2 (x) &#038;= \\int_1^N t^{x-1} e^{-t} dt , \\\\[6pt]<br \/>\n\\Gamma_3 (x) &#038;= \\int_N^\\infty t^{x-1} e^{-t} dt .<br \/>\n\\end{aligned}\\]<br \/>\n\\(0\\le t \\le 1\\)\uc77c \ub54c<br \/>\n\\[0\\le t^{x-1} e^{-t} \\le t^{x-1}\\]<br \/>\n\uc774\uace0,<br \/>\n\\[\\int_0^1 t^{x-1} dt = \\lim_{a\\rightarrow 0+} \\left[ \\frac{t^x}{x} \\right]_{t=a}^{t=1} = \\frac{1}{x}\\]<br \/>\n\uc774\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\Gamma_1 (x)\\)\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(\\Gamma_2 (x)\\)\ub294 \uc774\uc0c1\uc801\ubd84\uc774 \uc544\ub2cc \uc815\uc801\ubd84\uc774\ub2e4.<\/p>\n<p>\ub05d\uc73c\ub85c \\(\\Gamma_3 (x)\\)\uac00 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790. \\(t\\ge N\\)\uc77c \ub54c (46)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[0 \\le t^{x-1} e^{-t} \\le e^{-t\/2}\\]<br \/>\n\uc774\uace0,<br \/>\n\\[\\int_N^\\infty e^{-t\/2} dt = \\lim_{b\\rightarrow\\infty} \\bigg[ -2e^{-t\/2} \\bigg]_N^b = 2e^{-N\/2}\\]<br \/>\n\uc774\ubbc0\ub85c \ube44\uad50 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(\\Gamma_3 (x)\\)\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 \ub2e4\uc74c \uc0ac\uc2e4\uc744 \uc99d\uba85\ud558\uc600\ub2e4.<\/p>\n<div class=\"box\">\n<p>\uc784\uc758\uc758 \uc591\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(\\Gamma(x)\\)\uac00 \uc218\ub834\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc774\uc81c \uac10\ub9c8 \ud568\uc218\uac00 \ucc28\ub840\uacf1 \ud568\uc218\uc758 \uc131\uc9c8\uc744 \uac00\uc9c0\uace0 \uc788\uc74c\uc744 \ubcf4\uc774\uc790. \ubd80\ubd84\uc801\ubd84\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uba74<br \/>\n\\[\\begin{aligned}<br \/>\n\\Gamma (x+1)<br \/>\n&#038;= \\int_0^\\infty t^x e^{-t} dt \\\\[6pt]<br \/>\n&#038;= \\lim_{b\\rightarrow\\infty} \\left(<br \/>\n\\bigg[ -t^x e^{-t} \\bigg]_{t=0}^{t=b} + \\int_0^b xt^{x-1} e^{-t} dt<br \/>\n\\right) \\\\[6pt]<br \/>\n&#038;= 0+ x\\int_0^\\infty t^{x-1} e^{-x} dt \\\\[6pt]<br \/>\n&#038;= x\\Gamma(x)<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\Gamma(x+1) = x\\Gamma (x)\\tag{47}\\]<br \/>\n\uc774\ub2e4. \ub610\ud55c<br \/>\n\\[\\Gamma (1) = \\int_0^\\infty t^{1-1} e^{-t} dt = \\int_0^\\infty e^{-t} dt = 1\\tag{48}\\]<br \/>\n\uc774\ub2e4. \uc774\uc81c<br \/>\n\\[\\begin{aligned}<br \/>\n\\Gamma (2) &#038;= 1\\times \\Gamma (1) = 1\\times 1 = 1 = 1! ,\\\\[6pt]<br \/>\n\\Gamma (3) &#038;= 2\\times \\Gamma (2) = 2\\times 1! = 2! ,\\\\[6pt]<br \/>\n\\Gamma (4) &#038;= 3\\times \\Gamma (3) = 3\\times 2! = 3! ,\\\\[6pt]<br \/>\n\\,&#038;\\vdots<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\uba70, (47)\uacfc (48)\uc744 \uacb0\ud569\ud558\uace0 \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc744 \uc0ac\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"box\">\n<p>\\(0\\) \uc774\uc0c1\uc778 \uc784\uc758\uc758 \uc815\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\Gamma(n+1) = n!\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ########## ########## ########## ########## ########## --><br \/>\n<a name=\"outtro\"><\/a><\/p>\n<h3>\ub9fa\uc74c\ub9d0<\/h3>\n<p>\uc774 \uae00\uc5d0\uc11c\ub294 \uc774\uc0c1\uc801\ubd84\uc744 \uc815\uc758\ud558\uace0 \uc774\uc0c1\uc801\ubd84\uc758 \ud310\uc815\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc558\uc73c\uba70, \uc774\uc0c1\uc801\ubd84\uc744 \ud65c\uc6a9\ud558\ub294 \uc608\ub85c\uc11c \uac10\ub9c8 \ud568\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc558\ub2e4.<\/p>\n<p>\uc774 \uae00\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ud310\uc815\ubc95\uc740 \uc801\ubd84 \uad6c\uac04\uc774 \uc704\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0\uc640 \uc801\ubd84 \uad6c\uac04\uc758 \uc624\ub978\ucabd \ub05d\uc810\uc774 \ud2b9\uc774\uc810\uc778 \uacbd\uc6b0, \uadf8\ub9ac\uace0 \ud568\uc22b\uac12\uc774 \uc591\uc218\uc778 \uacbd\uc6b0\ub9cc \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \ud558\uc9c0\ub9cc \uc801\ubd84 \uad6c\uac04\uc774 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0, \uc801\ubd84 \uad6c\uac04\uc758 \uc67c\ucabd \ub05d\uc810\uc774\ub098 \uad6c\uac04\uc758 \uc548\ucabd\uc5d0 \uc788\ub294 \uc810\uc774 \ud2b9\uc774\uc810\uc778 \uacbd\uc6b0, \ud568\uc22b\uac12\uc774 \uc74c\uc218\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc774 \uae00\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ud310\uc815\ubc95\uacfc \ube44\uc2b7\ud55c \ud615\ud0dc\uc758 \ud310\uc815\ubc95\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\ud55c\ud3b8, \ubcc0\uc218\uac00 \ud558\ub098\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\uc758 \uc801\ubd84\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc911\uc801\ubd84, \uc120\uc801\ubd84, \uba74\uc801\ubd84\ub3c4 \uc774\uc0c1\uc801\ubd84\ub3c4 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc218\ud559\uc5d0 \uad00\uc2ec \uc788\ub294 \uc0ac\ub78c\uc774\ub77c\uba74 \uc774\uc640 \uac19\uc740 \ub0b4\uc6a9\uc744 \ucc3e\uc544\ubcf4\uae30 \ubc14\ub780\ub2e4.<\/p>\n<p><!--\n\u2018\u2019\n\u201c\u201d\n\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc5d0\uc11c\ub294 \ubcc0\uc218\uac00 \ud558\ub098\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\uc758 \uc774\uc0c1\uc801\ubd84\uc744 \uc815\uc758\ud558\uace0, \uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834 \ud310\uc815\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub610\ud55c \uc774\uc0c1\uc801\ubd84\uc744 \ud65c\uc6a9\ud558\ub294 \uc608\ub85c\uc11c \uac10\ub9c8 \ud568\uc218\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \ub0b4\uc6a9 \uc21c\uc11c \ub4e4\uc5b4\uac00\uae30 \uae38\uc774\uac00 \ubb34\ud55c\uc778 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\ub294 \uc774\uc0c1\uc801\ubd84 \uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\uc758 \uc774\uc0c1\uc801\ubd84 \uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834 \ud310\uc815\ubc95 (\uc801\ubd84 \uad6c\uac04\uc758 \uae38\uc774\uac00 \ubb34\ud55c\uc778 \uacbd\uc6b0) \uc774\uc0c1\uc801\ubd84\uc758 \uc218\ub834 \ud310\uc815\ubc95 (\ud568\uc218\uac00 \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0) \uc774\uc0c1\uc801\ubd84\uc744 \ud65c\uc6a9\ud558\ub294 \uc608: \uac10\ub9c8 \ud568\uc218 \ub9fa\uc74c\ub9d0 \ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9 \uc815\uc801\ubd84 (\uad00\ub828 \uae00) \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac (\uad00\ub828 \uae00) \ub4e4\uc5b4\uac00\uae30&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[350,579,576,578,580,342,574,575,339],"class_list":["post-8545","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-convergence-test","tag-gamma-function","tag-improper-integrals","tag-578","tag-580","tag-342","tag-574","tag-575","tag-339"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8545","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=8545"}],"version-history":[{"count":151,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8545\/revisions"}],"predecessor-version":[{"id":8720,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8545\/revisions\/8720"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=8545"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=8545"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=8545"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}