{"id":2823,"date":"2023-11-12T23:53:33","date_gmt":"2023-11-12T22:53:33","guid":{"rendered":"https:\/\/mxth.dk\/?p=2823"},"modified":"2026-04-09T22:41:56","modified_gmt":"2026-04-09T20:41:56","slug":"delvis-integration","status":"publish","type":"post","link":"https:\/\/mxth.dk\/?p=2823","title":{"rendered":"Delvis integration"},"content":{"rendered":"\n<p class=\" eplus-wrapper\"> Vi skal her se p\u00e5 delvis integration, som ogs\u00e5 kaldes for <em>partiel integration<\/em>, om er en metode vi kan benytte til at integrere integranter som best\u00e5r af produktet mellem to funktioner.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Selve regnereglen er som f\u00f8lger<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\int f(x)\\cdot g(x) dx = F(x)\\cdot g(x) &#8211; \\int F(x)\\cdot g\u2019(x) dx\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi vil starte med at se p\u00e5 et fire eksempler inden vi vil eftervise at formlen m\u00e5 g\u00e6lde.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi starter med et eksempel som bruger formlen direkte.<\/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\/1Pe_By5it8g?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\">Vi kan ogs\u00e5 v\u00e6re ude for at det andet integrale f\u00e5 for dannet ogs\u00e5 indeholder et produkt og her m\u00e5 vi alts\u00e5 endnu engang benytte formlen for delvis integration, vi skal alts\u00e5 g\u00f8re det to gange i tr\u00e6k. Men lad os se p\u00e5 eksemplet.<\/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\/SrFUINiaNZM?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\">i det tredje eksempel vil vi se p\u00e5 en integrant, hvor det ser ud til vi g\u00e5r lidt i ring, men hvor vi derved har mulighed for at g\u00f8re noget smart.<\/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\/8O_sLPJ7ogA?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\">Her til sidst vil vi se p\u00e5 et eksempel, hvor det ikke er nok at benytte delvis integration, men hvor vi ogs\u00e5 er n\u00f8d til at 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\/6Z1LjQfvAq0?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 at eftervise at formlen g\u00e6lder, har Stenner en glimrende video om kan tilg\u00e5es <a href=\"https:\/\/youtu.be\/NvaVBTk3Rdg?si=-vHi0YE_W7xhOUy_\" data-type=\"link\" data-id=\"https:\/\/youtu.be\/NvaVBTk3Rdg?si=-vHi0YE_W7xhOUy_\" target=\"_blank\" rel=\"noreferrer noopener\">her<\/a>.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Opgaver<\/strong><\/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:\/\/mxth.dk\/?page_id=920#IRDI001\" target=\"_blank\" rel=\"noopener noreferrer\">IRDI001<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=920#IRDI002\" target=\"_blank\" rel=\"noopener noreferrer\">IRDI002<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=920#IRDI003\" target=\"_blank\" rel=\"noopener noreferrer\">IRDI003<\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Vi skal her se p\u00e5 delvis integration, som ogs\u00e5 kaldes for partiel integration, om er en metode vi kan benytte til at integrere integranter som best\u00e5r af produktet mellem to funktioner. Selve regnereglen er som f\u00f8lger \\(\\int f(x)\\cdot g(x) dx = F(x)\\cdot g(x) &#8211; \\int F(x)\\cdot g\u2019(x) dx\\) Vi vil starte med at se p\u00e5 [&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_memberships_contains_paid_content":false,"footnotes":""},"categories":[13,3],"tags":[74],"class_list":["post-2823","post","type-post","status-publish","format-standard","hentry","category-integralregning","category-matematik","tag-htx"],"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\/2823","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=2823"}],"version-history":[{"count":4,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2823\/revisions"}],"predecessor-version":[{"id":2838,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2823\/revisions\/2838"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2823"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2823"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2823"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}