{"id":14,"date":"2022-08-18T17:17:18","date_gmt":"2022-08-18T17:17:18","guid":{"rendered":"https:\/\/www.prepaplus.tv\/blog\/?p=14"},"modified":"2022-08-18T17:17:18","modified_gmt":"2022-08-18T17:17:18","slug":"Comment-faire-face-a-une-integrale-impropre","status":"publish","type":"post","link":"https:\/\/www.prepaplus.tv\/blog\/Comment-faire-face-a-une-integrale-impropre\/","title":{"rendered":"Comment faire face \u00e0 une int\u00e9grale impropre ?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Essentiel au sein du programme de math\u00e9matiques en CPGE, il est important de savoir comment bien faire face \u00e0 une int\u00e9grale impropre ?&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Int\u00e9grale Impropre : Pourquoi se poser cette question existentielle ?&nbsp;<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Cette question n\u2019a l\u2019air de rien mais elle est fondamentale dans le programme de math\u00e9matiques du concours, en ECS, ECE, ECT ou m\u00eame ECG. En effet, <strong>le programme de math\u00e9matiques des concours s\u2019articule beaucoup autour des probabilit\u00e9s continues et variables al\u00e9atoires r\u00e9elles dite \u00ab&nbsp;a densit\u00e9&nbsp;\u00bb<\/strong>. Or, les densit\u00e9s de probabilit\u00e9s, les fonctions de r\u00e9partitions, les esp\u00e9rances, les variances, les moments (centr\u00e9s ou non), tous ces concepts sont en r\u00e9alit\u00e9 des int\u00e9grales impropres cach\u00e9es. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Il est donc essentiel de maitriser les int\u00e9grales impropres, et par maitriser on entend ici :<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n\n\n\n<li>Reconnaitre lorsqu\u2019une int\u00e9grale est impropre<\/li>\n\n\n\n\n\n\n<li>Savoir&nbsp;la calculer si n\u00e9cessaire<\/li>\n\n\n\n\n\n\n<li>Savoir la comparer \u00e0 une int\u00e9grale impropre \u00ab&nbsp;classique&nbsp;\u00bb du cours pour en d\u00e9terminer la nature (convergente ou divergente)<\/li>\n\n\n\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Les int\u00e9grales impropres jouent donc un r\u00f4le pr\u00e9pond\u00e9rant dans les sujets de maths au concours. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Quels sont les r\u00e9flexes \u00e0 avoir devant une int\u00e9grale impropre&nbsp;?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Lorsque l\u2019on fait face \u00e0 une int\u00e9grale impropre, il y a un certain plan d\u2019\u00e9tude \u00e0 respecter qui permet de ne pas faire d\u2019erreurs ou du moins de couvrir l\u2019ensemble des possibilit\u00e9s. Nous allons ici distinguer deux \u00e9tapes :&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Etape 1&nbsp;: Localiser \u00ab&nbsp;<em>l\u2019impropret\u00e9<\/em>&nbsp;\u00bb de l\u2019int\u00e9grale pour v\u00e9rifier si elle est (ou non) faussement impropre<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n\n\n\n<li>Si elle est impropre en un plus ou moins l\u2019infini, alors l\u2019int\u00e9grale ne peut pas \u00eatre faussement impropre<\/li>\n\n\n\n\n\n\n<li>Si elle est impropre en un r\u00e9el, alors il s\u2019agit de v\u00e9rifier si elle est faussement impropre.<\/li>\n\n\n\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Pour rappel :<\/strong>&nbsp;une int\u00e9grale est dite faussement impropre en un r\u00e9el \u00ab&nbsp;a&nbsp;\u00bb si la fonction int\u00e9gr\u00e9e est prolongeable par continuit\u00e9 en \u00ab&nbsp;a&nbsp;\u00bb. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>On v\u00e9rifie donc si la fonction est prolongeable en \u00ab&nbsp;a&nbsp;\u00bb par continuit\u00e9.<\/strong> Si tel est le cas, alors l\u2019int\u00e9grale n\u2019est plus impropre mais faussement impropre et on est donc ramen\u00e9 calcul d\u2019un int\u00e9grale finie, classique (en utilisant les m\u00e9thodes classiques&nbsp;: Int\u00e9gration a vue, Int\u00e9gration par parties, changement de variables).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Etape 2&nbsp;: Si l\u2019int\u00e9grale est toujours impropre, alors il s\u2019agit de d\u00e9terminer son comportement (convergente ou divergente) en la comparant \u00e0 une int\u00e9grale de Riemann (int\u00e9grale classique du cours)<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n\n\n\n<li>Si l\u2019int\u00e9grale diverge, alors l\u2019\u00e9tude s\u2019arr\u00eate<\/li>\n\n\n\n\n\n\n<li>Si l\u2019int\u00e9grale converge, alors on peut la calculer par les m\u00e9thodes classiques (int\u00e9gration a vue, Int\u00e9gration par parties, changement de variable)<\/li>\n\n\n\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Alors maintenant plus d&rsquo;excuse pour ne pas r\u00e9ussir \u00e0 bien manipuler une int\u00e9grale&nbsp;impropre le jour du concours !<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Essentiel au sein du programme de math\u00e9matiques en CPGE, il est important de savoir comment bien faire face \u00e0 une int\u00e9grale impropre ? <\/p>\n","protected":false},"author":2,"featured_media":83,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_kad_post_classname":"","footnotes":""},"categories":[5],"tags":[],"class_list":["post-14","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cours-maths-prepa-ecg"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Comment faire face \u00e0 une int\u00e9grale impropre ? - Pr\u00e9pa +<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.prepaplus.tv\/blog\/Comment-faire-face-a-une-integrale-impropre\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Comment faire face \u00e0 une int\u00e9grale impropre ? 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