{"id":1974,"date":"2019-04-08T12:15:06","date_gmt":"2019-04-08T03:15:06","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1974"},"modified":"2019-09-06T14:25:43","modified_gmt":"2019-09-06T05:25:43","slug":"calculus-antiderivatives","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-antiderivatives\/","title":{"rendered":"\uc5ed\ub3c4\ud568\uc218\uc640 \ubd80\uc815\uc801\ubd84"},"content":{"rendered":"<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubbf8\ubd84\uc758 \uc5ed\uacc4\uc0b0\uc778 \uc5ed\ub3c4\ud568\uc218\uc640 \ubd80\uc815\uc801\ubd84\uc758 \uac1c\ub150\uc744 \uc0b4\ud3b4\ubcf4\uace0, \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\ub294 \uba87 \uac00\uc9c0 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\uc5ed\ub3c4\ud568\uc218\uc640 \ubd80\uc815\uc801\ubd84<\/h3>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1.<\/span><br \/>\n\\(I\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(F\\)\uc640 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(I\\)\uc5d0\uc11c \\(F &#8216; = f\\)\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(F\\)\ub294 \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc5ed\ub3c4\ud568\uc218<\/span>(antiderivative)\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<\/div>\n<p>\ud55c \ud568\uc218\uc758 \uc5ed\ub3c4\ud568\uc218\ub294 \uc720\uc77c\ud558\uac8c \uc815\ud574\uc9c0\uc9c0 \uc54a\ub294\ub2e4. \uc608\ucee8\ub300<br \/>\n\\[f(x) = \\cos x\\]<br \/>\n\uc77c \ub54c, \ub2e4\uc74c \ud568\uc218\ub4e4\uc740 \ubaa8\ub450 \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc774\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nF_1 (x) &#038;= \\sin x ,\\\\[8pt]<br \/>\nF_2 (x) &#038;= \\sin x +4 ,\\\\[8pt]<br \/>\nF_3 (x) &#038;= \\sin x &#8211; \\pi .<br \/>\n\\end{align}\\]<br \/>\n\ud2b9\ud788 \ud3c9\uade0\uac12 \uc815\ub9ac\uc758 \ub530\ub984\uc815\ub9ac\ub85c\ubd80\ud130 \uc6b0\ub9ac\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc0ac\uc2e4\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span> \uad6c\uac04 \\(I\\)\uc5d0\uc11c \\(F\\)\uac00 \\(f\\)\uc758 \ud55c \uc5ed\ub3c4\ud568\uc218\uc774\uba74 \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \ubaa8\ub4e0 \uc5ed\ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\ub85c \ud45c\ud604\ub41c\ub2e4.<br \/>\n\\[F(x) +C\\]<br \/>\n\uc5ec\uae30\uc11c \\(C\\)\ub294 \uc784\uc758\uc758 \uc0c1\uc218\uc774\ub2e4.<\/p>\n<\/div>\n<p>\uc774\uc640 \uac19\uc740 \uad00\uc810\uc5d0\uc11c \ub2e4\uc74c \ub450 \ubb38\ud56d\uc740 \uc11c\ub85c \ub2e4\ub974\ub2e4.<\/p>\n<ul>\n<li>Find an antiderivative of \\(f\\) on \\(I .\\)<\/li>\n<li>Find the general antiderivative of \\(f\\) on \\(I .\\)<\/li>\n<\/ul>\n<p>\uccab \ubb38\ud56d\uc740 \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218 \uc911 \ud558\ub098\ub97c \uad6c\ud558\ub77c\ub294 \ub73b\uc774\uba70, \ub458\uc9f8 \ubb38\ud56d\uc740 \\(f\\)\uc774 \ubaa8\ub4e0 \uc5ed\ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub77c\ub294 \ub73b\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 1.<\/span><br \/>\nFind the general antiderivative of \\(f(x) = \\sin x\\) on \\(\\mathbb{R}.\\)\n<\/p>\n<p><span class=\"proof\">\ud480\uc774.<\/span> \\((-\\cos x)\\) is an antiderivative of \\(f(x)\\);<br \/>\n\\[\\frac{d}{dx} (-\\cos x ) = \\sin x .\\]<br \/>\nTherefore the general antiderivative formula for \\(f(x)\\) is<br \/>\n\\[F(x) = -\\cos x +C\\]<br \/>\nwhere \\(C\\) is an arbitrary constant.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<p>\uc8fc\uc5b4\uc9c4 \ud568\uc218\uc758 \ubaa8\ub4e0 \uc5ed\ub3c4\ud568\uc218\ub97c \ub098\ud0c0\ub0bc \ub54c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uae30\ud638\ub97c \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 2.<\/span><br \/>\n\ud568\uc218 \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218 \uc804\uccb4\ub97c \\(f\\)\uc758 <span class=\"defined\">\ubd80\uc815\uc801\ubd84<\/span>(indefinite integral)\uc774\ub77c\uace0 \ubd80\ub974\uba70 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\int f(x) dx \\tag{1}\\]<br \/>\n\uc5ec\uae30\uc11c \\(\\int\\)\ub97c <span class=\"defined\">\uc801\ubd84 \uae30\ud638<\/span>(integral sign)\ub77c\uace0 \ubd80\ub974\uace0 \\(f\\)\ub97c <span class=\"defined\">\ud53c\uc801\ubd84\ud568\uc218<\/span>(integrand)\ub77c\uace0 \ubd80\ub974\uba70 \\(x\\)\ub97c <span class=\"defined\">\uc801\ubd84\ubcc0\uc218<\/span>(variable of integration)\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/p>\n<\/div>\n<p>\uc5ed\ub3c4\ud568\uc218\uc758 \ubaa8\uc784\uc744 \u2018\ubd80\uc815\uc801\ubd84\u2019\uc774\ub77c\uace0 \ubd80\ub974\ub294 \uc774\uc720\ub294 \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\uc5d0\uc11c \uc815\uc801\ubd84\uc744 \uacc4\uc0b0\ud560 \ub54c \uc5ed\ub3c4\ud568\uc218\ub97c \uc0ac\uc6a9\ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span> \uc801\ubd84 \uae30\ud638 \uc0ac\uc6a9\ud558\uc5ec \ub098\ud0c0\ub0b4\uae30.<\/p>\n<p>\\[\\begin{align}<br \/>\n&#038;\\int 3x \\,dx = \\frac{3}{2} x^2 +C ,\\\\[4pt]<br \/>\n&#038;\\int \\sin x \\,dx = &#8211; \\cos x +C ,\\\\[4pt]<br \/>\n&#038;\\int \\left( e^t + \\sec^2 t + \\frac{1}{t} \\right) dt = e^t + \\tan t &#8211; \\frac{1}{t^2} +C .<br \/>\n\\end{align}\\]\n<\/p><\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\ucd08\uae43\uac12 \ubb38\uc81c<\/h3>\n<p>\ub4f1\uc2dd\uc774 \ud558\ub098 \uc774\uc0c1\uc758 \ubbf8\ubd84\uc5f0\uc0b0\uc790\ub97c \ud3ec\ud568\ud558\uace0 \uc788\uace0 \uadf8 \ub4f1\uc2dd\uc744 \ubbf8\ubd84\uc5f0\uc0b0\uc790\uac00 \ud3ec\ud568\ub418\uc9c0 \uc54a\uc740 \uc2dd\uc73c\ub85c \ud480\uc5b4\uc57c \ud560 \ub54c \uc6d0\ub798\uc758 \ub4f1\uc2dd\uc744 <span class=\"defined\">\ubbf8\ubd84\ubc29\uc815\uc2dd<\/span>(differential equation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc608\ucee8\ub300<br \/>\n\\[\\frac{dy}{dx} = \\sin x\\tag{2}\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(y=f(x)\\)\ub97c \ucc3e\ub294 \ubb38\uc81c\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \ub4f1\uc2dd (2)\ub97c \ubbf8\ubd84\ubc29\uc815\uc2dd\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. (2)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[f(x) = -\\cos x +C \\quad (C \\text{ is a constant})\\tag{3}\\]<br \/>\n\uac00 \ub41c\ub2e4. \uc774\ucc98\ub7fc \ubbf8\ubd84\ubc29\uc815\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub97c \uad6c\ud558\ub294 \uac83\uc744 <span class=\"defined\">\ubbf8\ubd84\ubc29\uc815\uc2dd\uc744 \ud47c\ub2e4<\/span>\ub77c\uace0 \ub9d0\ud558\uba70, \uad6c\ud55c \ud568\uc218\ub97c \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 <span class=\"defined\">\ud574<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 (3)\ucc98\ub7fc \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\uac00 \ub418\ub294 \ubaa8\ub4e0 \ud568\uc218\ub97c \uad6c\ud55c \uac83\uc744 <span class=\"defined\">\uc77c\ubc18\ud574<\/span>(general solution)\ub77c\uace0 \ubd80\ub974\uba70, \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574 \uc911 \ud558\ub098\ub97c \uad6c\ud55c \uac83\uc744 <span class=\"defined\">\ud2b9\uc218\ud574<\/span>(particular solution)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ub9cc\uc57d \ubbf8\ubd84\ubc29\uc815\uc2dd (2)\uc5d0<br \/>\n\\[f(0) = 3\\tag{4}\\]<br \/>\n\uc774\ub77c\ub294 \uc870\uac74\uc774 \ucd94\uac00\ub418\uba74 (2)\uc758 \ud574\ub294<br \/>\n\\[f(x) = -\\cos x +4 \\tag{5}\\]<br \/>\n\uac00 \ub41c\ub2e4. \uc774\ucc98\ub7fc \ubbf8\ubd84\ubc29\uc815\uc2dd\uc5d0\uc11c \\(x\\)\uc640 \\(y\\)\uc758 \uad00\uacc4\uc5d0 (4)\uc640 \uac19\uc740 \ud568\uc22b\uac12 \uc870\uac74\uc774 \ucd94\uac00\ub418\uc5c8\uc744 \ub54c, \ucd94\uac00\ub41c \uc870\uac74\uc744 <span class=\"defined\">\ucd08\uae43\uac12<\/span>(initial value)\uc774\ub77c\uace0 \ubd80\ub974\uba70, \ucd08\uae43\uac12 \uc870\uac74\uc774 \uc8fc\uc5b4\uc9c4 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc744 \ud478\ub294 \ubb38\uc81c\ub97c <span class=\"defined\">\ucd08\uae43\uac12 \ubb38\uc81c<\/span>(initial value problem)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 3.<\/span><br \/>\nSolve the initial value problem:<\/p>\n<p>\\[\\begin{align}<br \/>\n(1)\\,\\,&#038;\\frac{dy}{dx} = 2x-7 ,\\quad y(2)=0. \\\\[6pt]<br \/>\n(2)\\,\\,&#038;\\frac{ds}{dt} = 1+\\cos t ,\\quad s(\\pi )=1 . \\\\[6pt]<br \/>\n(3)\\,\\,&#038;\\frac{d^2 s}{dt^2} = \\frac{3t}{8} ;\\quad \\left.\\frac{ds}{dt}\\right\\vert_{t=4}=3 ,\\,\\, s(4)=4.<br \/>\n\\end{align}\\]<\/p>\n<p><span class=\"proof\">\ud480\uc774.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li style=\"text-align: left\">\\(y=x^2 -7x +C, \\) \\(\\,0=2^2 -14+C ,\\) \\(\\,C = 10,\\) \\(\\,\\,\\therefore \\,\\, y=x^2 -7x +10.\\)<\/li>\n<li style=\"text-align: left\">\\(s = t-\\sin t +C ,\\) \\(\\, 1 = \\pi &#8211; \\sin \\pi +C ,\\) \\(\\, C = 1-\\pi ,\\) \\(\\,\\, \\therefore \\,\\, s= t-\\sin t +1-\\pi .\\)<\/li>\n<li style=\"text-align: left\">\\(ds\/dt = 3t^2\/16 +C_1 ,\\) \\(\\, 3 = 3+C_1 ,\\) \\(\\, C_1 = 0 ,\\) \\(\\, ds\/dt = 3t^2  ,\\)<br \/>\n\\(\\,s= t^3 + C_2 ,\\) \\(\\,4=64+C_2 ,\\) \\(\\,C_2 = -60,\\) \\(\\,\\,\\therefore \\,\\, s=t^3 &#8211; 60.\\)\n<\/li>\n<\/ol>\n<\/div>\n<p><!--\n\n\n<div class=\"definition\">\n\n\n<p><span class=\"definition\">\uc815\uc758 n.<\/span>\n...\n<\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<p><!--\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\uc608\uc81c n.<\/span>\n...\n<\/p>\n\n\n\n\n\n<p><span class=\"proof\">\ud480\uc774.<\/span>\n...\n<span class=\"qee\"><\/span><\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<p><!--\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac n.<\/span><\/p>\n\n\n\n\n<p>..<\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p>...\n<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<p><!-- --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubbf8\ubd84\uc758 \uc5ed\uacc4\uc0b0\uc778 \uc5ed\ub3c4\ud568\uc218\uc640 \ubd80\uc815\uc801\ubd84\uc758 \uac1c\ub150\uc744 \uc0b4\ud3b4\ubcf4\uace0, \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\ub294 \uba87 \uac00\uc9c0 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc5ed\ub3c4\ud568\uc218\uc640 \ubd80\uc815\uc801\ubd84 \uc815\uc758 1. \\(I\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(F\\)\uc640 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(I\\)\uc5d0\uc11c \\(F &#8216; = f\\)\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(F\\)\ub294 \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218(antiderivative)\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ud55c \ud568\uc218\uc758 \uc5ed\ub3c4\ud568\uc218\ub294 \uc720\uc77c\ud558\uac8c \uc815\ud574\uc9c0\uc9c0 \uc54a\ub294\ub2e4. \uc608\ucee8\ub300 \\(f(x) = \\cos x\\) \uc77c \ub54c, \ub2e4\uc74c \ud568\uc218\ub4e4\uc740 \ubaa8\ub450 \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc774\ub2e4. \\(\\begin{align} F_1 (x) &#038;= \\sin x&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":[281,93,284,282,283,96,285,277,276,275,279,278,280],"class_list":["post-1974","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-antiderivative","tag-differential-equation","tag-general-solution","tag-indefinite-integral","tag-initial-value","tag-integral","tag-particular-solution","tag-277","tag-276","tag-275","tag-279","tag-278","tag-280"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1974","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=1974"}],"version-history":[{"count":33,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1974\/revisions"}],"predecessor-version":[{"id":3563,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1974\/revisions\/3563"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1974"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1974"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1974"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}