{"id":2810,"date":"2023-11-08T00:08:01","date_gmt":"2023-11-07T23:08:01","guid":{"rendered":"https:\/\/mxth.dk\/?p=2810"},"modified":"2023-11-12T20:30:33","modified_gmt":"2023-11-12T19:30:33","slug":"integration-ved-substitution","status":"publish","type":"post","link":"https:\/\/mxth.dk\/?p=2810","title":{"rendered":"Integration ved substitution"},"content":{"rendered":"\n<p class=\" eplus-wrapper\">Vi ser her lidt p\u00e5 integration ved substitution. Integration ved substitution er lidt lige som k\u00e6dereglen i differentialregningen, vi kan bruge den til at integrere sammensatte funktioner. Men hvor k\u00e6dereglen i differentialregningen at differentierer alle funktioner, kan integration ved substitution kun bruges i visse tilf\u00e6lde. Vi kommer til at se lidt p\u00e5 i hvilke tilf\u00e6lde det g\u00f8r sig g\u00e6ldende.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Lad os starte med at definere hvad integration ved substitution er<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(F(g(x))=\\int f(g(x))\\cdot g\u2019(x) dx\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Lad os se p\u00e5 et eksempel p\u00e5 hvordan man kan bruge denne formel.<\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\"><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/7SOFc9aHJmM?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Man kunne overveje hvad denne metode egentlig har med substitution af g\u00f8re s\u00e5 lad os se p\u00e5 en alternativ m\u00e5de<\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\"><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/IzvtzkpRY9Q?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Det er det samme som der sker men denne metode er lidt mere generel til at integrere integranter som er sammensatte. Lad os se p\u00e5 et par eksempler til<\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\"><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/jxQiE2O3XzU?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\"><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/VIdtFbNENxw?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\"><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/t7zn96kWyUI?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\">Det er dog ikke for alle integranter som indeholder sammensatte funktioner hvor vi kan bruge integration ved substitution<\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\"><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/wN0JPj3rO3A?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/p>\n\n\n\n<p class=\" eplus-wrapper\">Til sidst vil vi se p\u00e5 et eksempel, hvor vi ogs\u00e5 kan bruge integration ved substitution selvom der ikke er en sammensat funktion i integranten.<\/p>\n\n\n\n\n\n<p class=\" eplus-wrapper\"><span class=\"embed-youtube\" style=\"text-align:center; display: block;\"><iframe loading=\"lazy\" class=\"youtube-player\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/_MHr0qkNVuw?version=3&#038;rel=1&#038;showsearch=0&#038;showinfo=1&#038;iv_load_policy=1&#038;fs=1&#038;hl=da-DK&#038;autohide=2&#038;wmode=transparent\" allowfullscreen=\"true\" style=\"border:0;\" sandbox=\"allow-scripts allow-same-origin allow-popups allow-presentation allow-popups-to-escape-sandbox\"><\/iframe><\/span><\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/sites.google.com\/view\/matematikopgaverpmtekmatr4\/13-integralregning\/integration-ved-substitution\/opgave-389\" target=\"_blank\" rel=\"noopener noreferrer\">PM4-389<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/sites.google.com\/view\/matematikopgaverpmtekmatr4\/13-integralregning\/integration-ved-substitution\/opgave-390\" target=\"_blank\" rel=\"noopener noreferrer\">PM4-390<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxa3opgaver.systime.dk\/?id=233#c209\" target=\"_blank\" rel=\"noopener noreferrer\">matStxA3-1.47<\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/sites.google.com\/view\/matematikopgaverpmtekmatr4\/13-integralregning\/integration-ved-substitution\/opgave-391\" target=\"_blank\" rel=\"noopener noreferrer\">PM4-391<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxa3opgaver.systime.dk\/?id=233#c206\" target=\"_blank\" rel=\"noopener noreferrer\">matStxA3-1.45<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxa3opgaver.systime.dk\/?id=233#c216\" target=\"_blank\" rel=\"noopener noreferrer\">matStxA3-1.51<\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxa3opgaver.systime.dk\/?id=233#c203\" target=\"_blank\" rel=\"noopener noreferrer\">matStxA3-1.43<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" target=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" target=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxa3opgaver.systime.dk\/?id=233#c204\" target=\"_blank\" rel=\"noopener noreferrer\">matStxA3-1.44<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" target=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" target=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Vi ser her lidt p\u00e5 integration ved substitution. Integration ved substitution er lidt lige som k\u00e6dereglen i differentialregningen, vi kan bruge den til at integrere sammensatte funktioner. Men hvor k\u00e6dereglen i differentialregningen at differentierer alle funktioner, kan integration ved substitution kun bruges i visse tilf\u00e6lde. Vi kommer til at se lidt p\u00e5 i hvilke tilf\u00e6lde [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[13,3],"tags":[],"class_list":["post-2810","post","type-post","status-publish","format-standard","hentry","category-integralregning","category-matematik"],"featured_image_src":null,"author_info":{"display_name":"Henriksen","author_link":"https:\/\/mxth.dk\/?author=1"},"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2810","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2810"}],"version-history":[{"count":4,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2810\/revisions"}],"predecessor-version":[{"id":2815,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2810\/revisions\/2815"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2810"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2810"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2810"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}