{"id":1534,"date":"2026-09-08T08:08:39","date_gmt":"2026-09-08T08:08:39","guid":{"rendered":"https:\/\/maroczain.com\/scolaire.maroczain.com\/?page_id=1534"},"modified":"2026-09-08T08:08:39","modified_gmt":"2026-09-08T08:08:39","slug":"accumuler-transformer-calculer","status":"publish","type":"page","link":"https:\/\/maroczain.com\/scolaire.maroczain.com\/accumuler-transformer-calculer\/","title":{"rendered":"Accumuler. Transformer. Calculer"},"content":{"rendered":"\n&#8220;`html\n<div id=\"mza-integration-cpge\">\n\n<style>\n#mza-integration-cpge,#mza-integration-cpge *{box-sizing:border-box}\n#mza-integration-cpge{\n --navy:#06172c;--navy2:#0d416d;--gold:#c99a36;--gold2:#efd68c;\n --paper:#f5f6f8;--white:#fff;--text:#18283a;--muted:#687789;--line:#dfe5eb;\n font-family:Arial,Helvetica,sans-serif;background:var(--paper);color:var(--text);overflow:hidden\n}\n#mza-integration-cpge a{text-decoration:none;color:inherit}\n#mza-integration-cpge .wrap{max-width:1200px;margin:auto;padding:0 24px}\n#mza-integration-cpge section{padding:68px 0}\n#mza-integration-cpge .hero{\n color:#fff;padding:72px 0;\n background:\n radial-gradient(circle at 84% 15%,rgba(213,169,68,.28),transparent 28%),\n linear-gradient(135deg,#041224,#082b4c 65%,#10517e)\n}\n#mza-integration-cpge .back{display:inline-flex;color:#dbe6ee;font-size:12px;font-weight:900;margin-bottom:24px}\n#mza-integration-cpge .badge{\n display:inline-block;padding:8px 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.cta h2{color:#fff}\n#mza-integration-cpge .cta p{\n max-width:720px;margin:0 auto 22px;color:#cfdae4;line-height:1.7\n}\n@media(max-width:900px){\n #mza-integration-cpge .grid,#mza-integration-cpge .method{grid-template-columns:1fr}\n}\n<\/style>\n\n<header class=\"hero\">\n<div class=\"wrap\">\n\n<a class=\"back\" href=\"https:\/\/maroczain.com\/scolaire.maroczain.com\/mathematiques-cpge\/\">\n\u2190 Math\u00e9matiques CPGE\n<\/a>\n\n<br>\n<span class=\"badge\">COURS 07 \u2022 INT\u00c9GRATION<\/span>\n\n<h1>Accumuler.<br>Transformer. Calculer.<\/h1>\n\n<p class=\"lead\">\nL&#8217;int\u00e9gration est l&#8217;un des piliers de l&#8217;analyse en CPGE. Elle permet de calculer\ndes aires, des valeurs moyennes, des primitives, mais surtout de transformer\ndes expressions difficiles gr\u00e2ce au changement de variable et \u00e0 l&#8217;int\u00e9gration par parties.\n<\/p>\n\n<div class=\"actions\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda COMMENCER<\/a>\n<a href=\"#methode\" class=\"btn glass\">\ud83e\udde0 M\u00c9THODE<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 QCM<\/a>\n<\/div>\n\n<\/div>\n<\/header>\n\n<section id=\"cours\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">1. PRIMITIVES<\/span>\n<h2>Remonter de la d\u00e9riv\u00e9e vers la fonction.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>D\u00e9finition<\/h3>\n\n<p>\nUne fonction F est une primitive de f sur un intervalle I si :\n<\/p>\n\n<div class=\"formula\">\nF'(x)=f(x)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Famille des primitives<\/h3>\n\n<p>\nSi F est une primitive de f, alors toutes les primitives sont :\n<\/p>\n\n<div class=\"formula\">\nF(x)+C\n<\/div>\n\n<p>o\u00f9 C est une constante r\u00e9elle.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Primitive d&#8217;une puissance<\/h3>\n\n<div class=\"formula\">\n\u222b x\u207f dx = x\u207f\u207a\u00b9\/(n+1) + C\n<\/div>\n\n<p>pour n \u2260 -1.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Cas logarithmique<\/h3>\n\n<div class=\"formula\">\n\u222b 1\/x dx = ln|x| + C\n<\/div>\n\n<p>sur tout intervalle ne contenant pas 0.<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">2. INT\u00c9GRALE SUR UN SEGMENT<\/span>\n<h2>Passer de la primitive \u00e0 l&#8217;accumulation.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Th\u00e9or\u00e8me fondamental<\/h3>\n\n<p>\nSi f est continue sur [a,b] et F est une primitive de f :\n<\/p>\n\n<div class=\"formula\">\n\u222b\u2090\u1d47 f(x) dx = F(b)-F(a)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Orientation<\/h3>\n\n<div class=\"formula\">\n\u222b\u2090\u1d47 f(x)dx = -\u222b\u1d66\u1d43 f(x)dx\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Additivit\u00e9<\/h3>\n\n<div class=\"formula\">\n\u222b\u2090\u1d47 f + \u222b\u1d66\u1d9c f = \u222b\u2090\u1d9c f\n<\/div>\n\n<p>\nC&#8217;est la relation de Chasles.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Lin\u00e9arit\u00e9<\/h3>\n\n<div class=\"formula\">\n\u222b(\u03b1f+\u03b2g)\n=\n\u03b1\u222bf + \u03b2\u222bg\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">3. POSITIVIT\u00c9 &#038; COMPARAISON<\/span>\n<h2>Encadrer sans calculer exactement.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Positivit\u00e9<\/h3>\n\n<p>\nSi f(x) \u2265 0 sur [a,b], alors :\n<\/p>\n\n<div class=\"formula\">\n\u222b\u2090\u1d47 f(x)dx \u2265 0\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Comparaison<\/h3>\n\n<p>\nSi :\n<\/p>\n\n<div class=\"formula\">\nf(x) \u2264 g(x)\n<\/div>\n\n<p>sur [a,b], alors :<\/p>\n\n<div class=\"formula\">\n\u222b\u2090\u1d47 f(x)dx \u2264 \u222b\u2090\u1d47 g(x)dx\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Valeur absolue<\/h3>\n\n<div class=\"formula\">\n|\u222b\u2090\u1d47 f(x)dx|\n\u2264\n\u222b\u2090\u1d47 |f(x)|dx\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Encadrement simple<\/h3>\n\n<p>\nSi m \u2264 f(x) \u2264 M sur [a,b], alors :\n<\/p>\n\n<div class=\"formula\">\nm(b-a)\n\u2264\n\u222b\u2090\u1d47 f(x)dx\n\u2264\nM(b-a)\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"methode\" class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA INTEGRAL ENGINE<\/span>\n<h2>La strat\u00e9gie en six r\u00e9flexes.<\/h2>\n<\/div>\n\n<div class=\"method\">\n<div><b>01<\/b><span>SIMPLIFIER<\/span><\/div>\n<div><b>02<\/b><span>RECONNA\u00ceTRE<\/span><\/div>\n<div><b>03<\/b><span>PRIMITIVE DIRECTE<\/span><\/div>\n<div><b>04<\/b><span>CHANGEMENT VARIABLE<\/span><\/div>\n<div><b>05<\/b><span>IPP<\/span><\/div>\n<div><b>06<\/b><span>V\u00c9RIFIER<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">4. CHANGEMENT DE VARIABLE<\/span>\n<h2>Transformer l&#8217;int\u00e9grale.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Principe<\/h3>\n\n<p>\nOn pose :\n<\/p>\n\n<div class=\"formula\">\nx = \u03c6(t)\n<\/div>\n\n<p>alors :<\/p>\n\n<div class=\"formula\">\ndx = \u03c6'(t)dt\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Formule<\/h3>\n\n<div class=\"formula\">\n\u222b\u2090\u1d47 f(x)dx\n=\n\u222b\u03b1\u1d5d f(\u03c6(t))\u03c6'(t)dt\n<\/div>\n\n<p>\no\u00f9 \u03c6(\u03b1)=a et \u03c6(\u03b2)=b.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Quand l&#8217;utiliser ?<\/h3>\n\n<ul>\n<li>composition \u00e9vidente ;<\/li>\n<li>racine ;<\/li>\n<li>fonction rationnelle ;<\/li>\n<li>sym\u00e9trie ;<\/li>\n<li>expression trigonom\u00e9trique.<\/li>\n<\/ul>\n<\/div>\n\n<div class=\"card\">\n<h3>R\u00e9flexe important<\/h3>\n\n<div class=\"tip\">\nDans une int\u00e9grale d\u00e9finie, modifiez aussi les bornes lorsque vous changez de variable.\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">5. INT\u00c9GRATION PAR PARTIES<\/span>\n<h2>D\u00e9placer une d\u00e9riv\u00e9e.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nPour deux fonctions suffisamment r\u00e9guli\u00e8res :\n<\/p>\n\n<div class=\"formula\">\n\u222b\u2090\u1d47 u(x)v'(x)dx\n=\n[u(x)v(x)]\u2090\u1d47\n&#8211;\n\u222b\u2090\u1d47 u'(x)v(x)dx\n<\/div>\n\n<div class=\"tip\">\nLa bonne int\u00e9gration par parties est celle qui produit une int\u00e9grale plus simple que l&#8217;int\u00e9grale de d\u00e9part.\n<\/div>\n\n<\/div>\n\n<div class=\"grid\" style=\"margin-top:16px\">\n\n<div class=\"card\">\n<h3>Cas typique<\/h3>\n\n<div class=\"formula\">\n\u222b x e\u02e3 dx\n<\/div>\n\n<p>\nOn d\u00e9rive le facteur polynomial et on primitive l&#8217;exponentielle.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Logarithme<\/h3>\n\n<div class=\"formula\">\n\u222b ln(x) dx\n<\/div>\n\n<p>\nOn peut consid\u00e9rer ln(x) \u00d7 1.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">6. SYM\u00c9TRIES UTILES<\/span>\n<h2>R\u00e9duire le calcul gr\u00e2ce \u00e0 la structure.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Fonction paire<\/h3>\n\n<p>\nSi f est paire :\n<\/p>\n\n<div class=\"formula\">\n\u222b\u208b\u2090\u1d43 f(x)dx\n=\n2\u222b\u2080\u1d43 f(x)dx\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Fonction impaire<\/h3>\n\n<p>\nSi f est impaire :\n<\/p>\n\n<div class=\"formula\">\n\u222b\u208b\u2090\u1d43 f(x)dx = 0\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Transformation x \u2192 a+b-x<\/h3>\n\n<p>\nElle peut r\u00e9v\u00e9ler une sym\u00e9trie cach\u00e9e dans une int\u00e9grale sur [a,b].\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>R\u00e9flexe concours<\/h3>\n\n<div class=\"tip\">\nAvant de lancer des calculs longs, cherchez toujours une sym\u00e9trie ou une substitution naturelle.\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">7. INT\u00c9GRALES IMPROPRES<\/span>\n<h2>Quand une borne ou l&#8217;int\u00e9grande devient probl\u00e9matique.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Borne infinie<\/h3>\n\n<p>\nOn d\u00e9finit par exemple :\n<\/p>\n\n<div class=\"formula\">\n\u222b\u2090\u207a\u221e f(x)dx\n=\nlim \u222b\u2090\u1d2c f(x)dx\n<br>\nA \u2192 +\u221e\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Singularit\u00e9 en une borne<\/h3>\n\n<p>\nSi f devient non born\u00e9e au voisinage de a :\n<\/p>\n\n<div class=\"formula\">\n\u222b\u2090\u1d47 f(x)dx\n=\nlim \u222b\u209c\u1d47 f(x)dx\n<br>\nt \u2192 a\u207a\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Convergence<\/h3>\n\n<p>\nL&#8217;int\u00e9grale impropre converge si la limite d\u00e9finissant l&#8217;int\u00e9grale existe et est finie.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Int\u00e9grale de r\u00e9f\u00e9rence<\/h3>\n\n<div class=\"formula\">\n\u222b\u2081\u207a\u221e 1\/x\u1d56 dx\n<\/div>\n\n<p>\nSa convergence d\u00e9pend de la valeur de p.\n<\/p>\n\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"exercices\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">8. EXERCICES PROGRESSIFS<\/span>\n<h2>Choisir la bonne technique.<\/h2>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 1<\/span>\n<h3>Exercice 1 \u2014 Primitive<\/h3>\n\n<p>D\u00e9terminer une primitive de :<\/p>\n\n<div class=\"formula\">\nf(x)=3x\u00b2+2x-1\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaIntHint('intHint1')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"intHint1\" class=\"hint\">\nPrimitivez chaque terme s\u00e9par\u00e9ment.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 2<\/span>\n<h3>Exercice 2 \u2014 Changement de variable<\/h3>\n\n<p>Calculer :<\/p>\n\n<div class=\"formula\">\n\u222b\u2080\u00b9 2x e<sup>x\u00b2<\/sup> dx\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaIntHint('intHint2')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"intHint2\" class=\"hint\">\nLa pr\u00e9sence de 2x \u00e0 c\u00f4t\u00e9 de x\u00b2 sugg\u00e8re une substitution tr\u00e8s directe.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 3<\/span>\n<h3>Exercice 3 \u2014 Int\u00e9gration par parties<\/h3>\n\n<p>Calculer :<\/p>\n\n<div class=\"formula\">\n\u222b\u2080\u00b9 x e\u02e3 dx\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaIntHint('intHint3')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"intHint3\" class=\"hint\">\nChoisissez u=x et v&#8217;=e\u02e3.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU PR\u00c9PA<\/span>\n<h3>Exercice 4 \u2014 Sym\u00e9trie<\/h3>\n\n<p>\nOn pose :\n<\/p>\n\n<div class=\"formula\">\nI = \u222b\u2080\u00b9 ln(1+x)\/(1+x\u00b2) dx\n<\/div>\n\n<p>\nCherchez une transformation de variable susceptible de faire appara\u00eetre une relation entre deux expressions de I.\n<\/p>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaIntHint('intHint4')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"intHint4\" class=\"hint\">\nDans les probl\u00e8mes d&#8217;int\u00e9grales sur [0,1], la transformation x \u2192 1\/x appara\u00eet souvent apr\u00e8s adaptation des bornes, mais d&#8217;autres sym\u00e9tries peuvent \u00eatre plus naturelles selon la structure.\n<\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">9. PROBLEM LAB<\/span>\n<h2>Mission : construire une relation de r\u00e9currence.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<h3>Int\u00e9grales de Wallis<\/h3>\n\n<p>\nPour n entier naturel, on pose :\n<\/p>\n\n<div class=\"formula\">\nI\u2099 = \u222b\u2080<sup>\u03c0\/2<\/sup> sin\u207f(x) dx\n<\/div>\n\n<p>Votre mission :<\/p>\n\n<ol>\n<li>calculer I\u2080 et I\u2081 ;<\/li>\n<li>utiliser une int\u00e9gration par parties pour relier I\u2099 \u00e0 I\u2099\u208b\u2082 ;<\/li>\n<li>\u00e9tudier le comportement de la suite (I\u2099) ;<\/li>\n<li>comprendre comment cette famille d&#8217;int\u00e9grales peut servir \u00e0 obtenir des estimations asymptotiques.<\/li>\n<\/ol>\n\n<div class=\"actions\">\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaIntHint('intMission1')\">\ud83d\udca1 INDICE<\/button>\n<button class=\"btn\" style=\"background:#06172c;color:#fff\" type=\"button\" onclick=\"mzaIntHint('intMission2')\">\ud83e\udde0 M\u00c9THODE<\/button>\n<\/div>\n\n<div id=\"intMission1\" class=\"hint\">\n\u00c9crivez sin\u207f(x)=sin\u207f\u207b\u00b9(x)\u00b7sin(x), puis choisissez soigneusement les deux facteurs pour l&#8217;int\u00e9gration par parties.\n<\/div>\n\n<div id=\"intMission2\" class=\"hint\">\nApr\u00e8s l&#8217;int\u00e9gration par parties, utilisez cos\u00b2(x)=1-sin\u00b2(x) pour faire r\u00e9appara\u00eetre des int\u00e9grales de la m\u00eame famille.\nNe cherchez pas \u00e0 tout calculer directement.\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"qcm\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">10. QCM DE VALIDATION<\/span>\n<h2>Diagnostic Int\u00e9gration CPGE.<\/h2>\n<p>Le score est affich\u00e9 sans r\u00e9v\u00e9ler les bonnes r\u00e9ponses.<\/p>\n<\/div>\n\n<div class=\"qcm\">\n\n<form id=\"integrationQuiz\">\n\n<div class=\"question\">\n<strong>1. Si F est une primitive de f, alors :<\/strong>\n<label><input type=\"radio\" name=\"i1\" value=\"1\"> F&#8217;=f<\/label>\n<label><input type=\"radio\" name=\"i1\" value=\"0\"> F=f&#8217;<\/label>\n<label><input type=\"radio\" name=\"i1\" value=\"0\"> F=f\u00b2<\/label>\n<label><input type=\"radio\" name=\"i1\" value=\"0\"> F est toujours constante<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>2. Pour une fonction continue sur [a,b] :<\/strong>\n<label><input type=\"radio\" name=\"i2\" value=\"1\"> \u222b\u2090\u1d47 f = F(b)-F(a)<\/label>\n<label><input type=\"radio\" name=\"i2\" value=\"0\"> \u222b\u2090\u1d47 f = F(a)+F(b)<\/label>\n<label><input type=\"radio\" name=\"i2\" value=\"0\"> \u222b\u2090\u1d47 f = f(b)-f(a)<\/label>\n<label><input type=\"radio\" name=\"i2\" value=\"0\"> l&#8217;int\u00e9grale est toujours nulle<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>3. Si f est impaire sur [-a,a], alors :<\/strong>\n<label><input type=\"radio\" name=\"i3\" value=\"1\"> son int\u00e9grale sur [-a,a] est nulle<\/label>\n<label><input type=\"radio\" name=\"i3\" value=\"0\"> son int\u00e9grale est toujours positive<\/label>\n<label><input type=\"radio\" name=\"i3\" value=\"0\"> son int\u00e9grale vaut 2f(a)<\/label>\n<label><input type=\"radio\" name=\"i3\" value=\"0\"> elle est forc\u00e9ment constante<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>4. Lors d&#8217;un changement de variable dans une int\u00e9grale d\u00e9finie :<\/strong>\n<label><input type=\"radio\" name=\"i4\" value=\"1\"> il faut adapter aussi les bornes<\/label>\n<label><input type=\"radio\" name=\"i4\" value=\"0\"> les bornes ne changent jamais<\/label>\n<label><input type=\"radio\" name=\"i4\" value=\"0\"> on supprime toujours dx<\/label>\n<label><input type=\"radio\" name=\"i4\" value=\"0\"> on d\u00e9rive l&#8217;int\u00e9grale enti\u00e8re<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>5. L&#8217;int\u00e9gration par parties sert principalement \u00e0 :<\/strong>\n<label><input type=\"radio\" name=\"i5\" value=\"1\"> transformer une int\u00e9grale en une autre plus simple<\/label>\n<label><input type=\"radio\" name=\"i5\" value=\"0\"> calculer uniquement des polyn\u00f4mes<\/label>\n<label><input type=\"radio\" name=\"i5\" value=\"0\"> trouver les racines d&#8217;un polyn\u00f4me<\/label>\n<label><input type=\"radio\" name=\"i5\" value=\"0\"> prouver automatiquement la convergence<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>6. Une int\u00e9grale impropre est convergente lorsque :<\/strong>\n<label><input type=\"radio\" name=\"i6\" value=\"1\"> la limite qui la d\u00e9finit existe et est finie<\/label>\n<label><input type=\"radio\" name=\"i6\" value=\"0\"> son int\u00e9grande est toujours positive<\/label>\n<label><input type=\"radio\" name=\"i6\" value=\"0\"> la borne sup\u00e9rieure est infinie<\/label>\n<label><input type=\"radio\" name=\"i6\" value=\"0\"> elle contient un logarithme<\/label>\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaIntegrationScore()\">\nVALIDER MON QCM\n<\/button>\n\n<\/form>\n\n<div id=\"integrationResult\"><\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"cta\">\n<div style=\"font-size:43px\">\u222b<\/div>\n\n<h2>Int\u00e9grer, c&#8217;est reconna\u00eetre avant de calculer.<\/h2>\n\n<p>\nUne int\u00e9grale difficile devient souvent simple d\u00e8s que l&#8217;on identifie\nla bonne structure : primitive directe, sym\u00e9trie, substitution,\nint\u00e9gration par parties ou comparaison.\n<\/p>\n\n<div class=\"actions\" style=\"justify-content:center\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda REVOIR LE COURS<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 QCM<\/a>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<script>\nfunction mzaIntHint(id){\n const el=document.getElementById(id);\n if(!el)return;\n el.style.display = el.style.display===\"block\" ? \"none\" : \"block\";\n}\n\nfunction mzaIntegrationScore(){\n const form=document.getElementById(\"integrationQuiz\");\n const result=document.getElementById(\"integrationResult\");\n let score=0;\n let complete=true;\n\n [\"i1\",\"i2\",\"i3\",\"i4\",\"i5\",\"i6\"].forEach(function(name){\n   const answer=form.querySelector('input[name=\"'+name+'\"]:checked');\n   if(!answer){complete=false}\n   else{score+=Number(answer.value)}\n });\n\n result.style.display=\"block\";\n\n if(!complete){\n   result.innerHTML=\"<strong>R\u00e9pondez aux 6 questions avant de valider.<\/strong>\";\n   return;\n }\n\n const pct=Math.round(score\/6*100);\n\n let level=\"Int\u00e9gration \u00e0 consolider\";\n if(pct>=50) level=\"Bases acquises\";\n if(pct>=67) level=\"Bon niveau\";\n if(pct>=84) level=\"Tr\u00e8s bonne ma\u00eetrise\";\n if(pct===100) level=\"Excellent niveau CPGE\";\n\n result.innerHTML=\n \"<strong style='font-size:30px;color:#efd68c'>\"+pct+\"%<\/strong>\"+\n \"<p><b>\"+level+\"<\/b><\/p>\"+\n \"<p>Score : \"+score+\" \/ 6<\/p>\"+\n \"<p>Les r\u00e9ponses correctes et la correction d\u00e9taill\u00e9e restent masqu\u00e9es.<\/p>\";\n}\n<\/script>\n\n<\/div>\n&#8220;`\n","protected":false},"excerpt":{"rendered":"<p>&#8220;`html \u2190 Math\u00e9matiques CPGE COURS 07 \u2022 INT\u00c9GRATION Accumuler.Transformer. Calculer. L&#8217;int\u00e9gration est l&#8217;un des piliers de l&#8217;analyse en CPGE. Elle permet de calculer des aires, des valeurs moyennes, des primitives, mais surtout de transformer des expressions difficiles gr\u00e2ce au changement de variable et \u00e0 l&#8217;int\u00e9gration par parties. \ud83d\udcda COMMENCER \ud83e\udde0 M\u00c9THODE \ud83e\udde9 EXERCICES \u2753 QCM [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1534","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1534","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/comments?post=1534"}],"version-history":[{"count":1,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1534\/revisions"}],"predecessor-version":[{"id":1536,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1534\/revisions\/1536"}],"wp:attachment":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/media?parent=1534"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}