{"id":32670,"date":"2023-04-04T13:00:37","date_gmt":"2023-04-04T13:00:37","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32670"},"modified":"2025-03-27T12:31:06","modified_gmt":"2025-03-27T12:31:06","slug":"integrales-indefinies-et-techniques-de-base-dintegration","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/integrales-indefinies-et-techniques-de-base-dintegration\/","title":{"rendered":"Int\u00e9grales Ind\u00e9finies et Techniques de Base d\u2019Int\u00e9gration"},"content":{"rendered":"<style>\n    p, ul, ol {\n        text-align: justify;\n    }\n    h1, h2, h3 {\n    text-align:center;\n    }\n<\/style>\n<p><center><\/p>\n<h1>Int\u00e9grales Ind\u00e9finies et Techniques de Base d\u2019Int\u00e9gration<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\">Dans ce cours, les techniques de base pour calculer les int\u00e9grales ind\u00e9finies les plus \u00e9l\u00e9mentaires sont introduites, ainsi que les propri\u00e9t\u00e9s de l\u2019op\u00e9rateur d\u2019int\u00e9gration. Cela comprend les int\u00e9grales polynomiales, exponentielles, hyperboliques et trigonom\u00e9triques de base.<\/p>\n<p style=\"text-align:center;\"><strong><u>Objectifs d\u2019Apprentissage<\/u> :<\/strong><br \/>\u00c0 la fin de ce cours, l\u2019\u00e9tudiant sera capable de<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> le processus d\u2019int\u00e9gration ind\u00e9finie comme le processus inverse de la d\u00e9rivation.<\/li>\n<li><strong>Calculer<\/strong> l\u2019int\u00e9grale de polyn\u00f4mes et d\u2019expressions impliquant des fonctions exponentielles, hyperboliques et trigonom\u00e9triques.<\/li>\n<li><strong>Utiliser<\/strong> les propri\u00e9t\u00e9s des int\u00e9grales pour effectuer des manipulations alg\u00e9briques facilitant leur calcul.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>TABLE DES MATI\u00c8RES<\/strong><br \/>\n<a href=\"#1\">LA PERTINENCE DES INT\u00c9GRALES IND\u00c9FINIES<\/a><br \/>\n<a href=\"#2\">ANTID\u00c9RIV\u00c9ES, INT\u00c9GRALES IND\u00c9FINIES ET PRIMITIVES DE FONCTIONS<\/a><br \/>\n<a href=\"#3\">TECHNIQUES DE BASE D\u2019INT\u00c9GRATION<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/4wSTxA7zY9k\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>La pertinence des int\u00e9grales ind\u00e9finies<\/h2>\n<p>Les int\u00e9grales ind\u00e9finies sont un outil fondamental en calcul et ont une large gamme d\u2019applications dans les sciences physiques et math\u00e9matiques. Elles permettent de calculer la fonction primitive d\u2019une fonction donn\u00e9e, ce qui est ensuite utilis\u00e9 pour calculer des aires sous des courbes, des volumes de solides, des probabilit\u00e9s et de nombreuses autres applications en physique, ing\u00e9nierie, statistique et \u00e9conomie. De plus, les int\u00e9grales ind\u00e9finies sont essentielles pour la r\u00e9solution d\u2019\u00e9quations diff\u00e9rentielles, ce qui les rend indispensables dans de nombreux domaines scientifiques et technologiques.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/56fMLiVPwDI\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"2\"><\/a><\/p>\n<h2>Antid\u00e9riv\u00e9es, int\u00e9grales ind\u00e9finies et primitives de fonctions<\/h2>\n<p>Si une fonction <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> a pour d\u00e9riv\u00e9e <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> sur un intervalle donn\u00e9 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>, alors on dit que <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> est une primitive de <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> sur cet intervalle.<\/p>\n<p>Il est important de noter que si <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> est une primitive de <span class=\"katex-eq\" data-katex-display=\"false\">f(x),<\/span> alors <span class=\"katex-eq\" data-katex-display=\"false\">F(x) + C<\/span> l\u2019est \u00e9galement, o\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est une constante r\u00e9elle quelconque. Cela s\u2019exprime ainsi :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int f(x) dx = F(x) + C<\/span>\n<p>La constante <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est ce qu\u2019on appelle une <strong>constante d\u2019int\u00e9gration<\/strong>, et sa pr\u00e9sence indique que la primitive d\u2019une fonction n\u2019est pas une fonction unique, mais une famille de fonctions : l\u2019ensemble de toutes les fonctions dont la d\u00e9riv\u00e9e est <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> sur l\u2019intervalle <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>.<\/p>\n<p>Les termes antid\u00e9riv\u00e9e, primitive et int\u00e9grale ind\u00e9finie sont trois fa\u00e7ons d\u2019exprimer la m\u00eame id\u00e9e, et sont donc utilis\u00e9s de mani\u00e8re interchangeable. En r\u00e9sum\u00e9, l\u2019int\u00e9grale ind\u00e9finie est le processus inverse du calcul des d\u00e9riv\u00e9es, et c\u2019est \u00e0 partir de cette id\u00e9e que ses propri\u00e9t\u00e9s fondamentales sont obtenues.<\/p>\n<h3>Propri\u00e9t\u00e9s de base des int\u00e9grales ind\u00e9finies<\/h3>\n<p>Pour pouvoir calculer les int\u00e9grales ind\u00e9finies, nous devons d\u2019abord conna\u00eetre certaines propri\u00e9t\u00e9s de base, qui sont directement h\u00e9rit\u00e9es des propri\u00e9t\u00e9s des d\u00e9riv\u00e9es.<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int  \\dfrac{df(x)}{dx} dx = f(x) + C<\/span><\/br>Parce que l\u2019int\u00e9grale ind\u00e9finie est le processus inverse de la d\u00e9rivation.<\/li>\n<p><\/br><\/p>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\lambda f(x) dx = \\lambda \\int f(x) dx<\/span><\/br>O\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda<\/span> est une constante r\u00e9elle quelconque. Cela est vrai parce que<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int \\lambda \\dfrac{d\\phi(x)}{dx}dx &amp;=  \\displaystyle \\int \\dfrac{d}{dx}\\lambda \\phi(x) dx \\\\ \\\\\n\n&amp;= \\lambda \\phi(x) + C_1 \\\\ \\\\\n\n&amp;= \\lambda(\\phi(x) + C_2) \\\\ \\\\\n\n&amp;= \\lambda \\displaystyle  \\int \\frac{d\\phi(x)}{dx}dx \\end{array}<\/span><\/center><br \/>\n<\/br><br \/>\nEt ensuite, en utilisant <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{d\\phi(x)}{dx}<\/span>, on obtient<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\lambda f(x) dx = \\lambda \\int f(x)dx<\/span><\/center><\/li>\n<p><\/br><\/p>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int f(x) + g(x) dx = \\int f(x) dx + \\int g(x) dx <\/span>\n<\/br><br \/>\nCela peut \u00eatre d\u00e9montr\u00e9 de fa\u00e7on similaire \u00e0 l\u2019exemple pr\u00e9c\u00e9dent. Consid\u00e9rons deux fonctions <span class=\"katex-eq\" data-katex-display=\"false\">\\phi(x)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\psi(x)<\/span> telles que<br \/>\n<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{d\\phi(x)}{dx}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\dfrac{d\\psi(x)}{dx}<\/span><\/center><br \/>\n<\/br><br \/>\nAlors nous avons<br \/>\n<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int f(x) + g(x) dx\n\n&amp;= \\displaystyle \\int \\dfrac{d\\phi(x)}{dx} +  \\dfrac{d\\psi(x)}{dx} dx \\\\ \\\\\n\n&amp;= \\displaystyle \\int \\dfrac{d}{dx} (\\phi(x)  + \\psi(x)) dx \\\\ \\\\\n\n&amp;= \\phi(x) + \\psi(x) + C \\\\ \\\\\n\n&amp;= (\\phi(x) + C_1) + (\\psi(x) + C_2) \\\\ \\\\\n\n&amp;= \\displaystyle \\int \\dfrac{d\\phi(x)}{dx} dx + \\int \\dfrac{d\\psi(x)}{dx}dx \\\\ \\\\\n\n&amp;= \\displaystyle \\int f(x) dx + \\int g(x) dx\n\n\\end{array}<\/span><\/center>\n<\/li>\n<\/ol>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Techniques de base d\u2019int\u00e9gration<\/h2>\n<p>Il existe des techniques de base d\u2019int\u00e9gration qui nous permettent de calculer certaines int\u00e9grales ind\u00e9finies \u00e0 partir des r\u00e9sultats obtenus par d\u00e9rivation. Gr\u00e2ce \u00e0 ces techniques, nous pouvons obtenir les r\u00e9sultats suivants utiles \u00e0 l\u2019int\u00e9gration :<\/p>\n<h3>Int\u00e9grales de fonctions polynomiales<\/h3>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int 1 dx = x + C<\/span>\n<\/br><br \/>\nParce que <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} (x + C)= 1 <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int x^q dx = \\dfrac{x^{q+1}}{q+1}  + C,<\/span> \u00e0 condition que <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq -1<\/span>\n<\/br><br \/>\nParce que <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left(\\dfrac{x^{q+1}}{q+1}  + C\\right) = x^q.<\/span>\n<\/li>\n<\/ol>\n<p>Avec ces r\u00e9sultats et les propri\u00e9t\u00e9s de base, nous pouvons calculer sans difficult\u00e9 l\u2019int\u00e9grale de tout polyn\u00f4me.<\/p>\n<div style=\"background-color:#F3FFF3; padding:20px;\">\n<p><strong>Exemple :<\/strong><\/p>\n<ol>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 3x+2 \\right) dx =  \\dfrac{3}{2}x^2 + 2x + C<\/span><\/li>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 5x^2 + 2x + 3 \\right) dx= \\dfrac{5}{3}x^3 + x + 3x  + C<\/span><\/li>\n<li type=\"a\"> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 4x^{12} - 7x^{-1\/3} + 1 \\right) dx  <\/span> <\/li>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} &amp;= \\dfrac{4}{13}x^{13} - \\dfrac{7}{2\/3}x^{2\/3} + x + C \\\\ \\\\\n\n&amp;= \\dfrac{4}{13}x^{13} - \\dfrac{21}{2}x^{2\/3} + x + C\n\n\\end{array}<\/span>\n<\/ol>\n<\/div>\n<h3>Int\u00e9grales de fonctions exponentielles et logarithmiques<\/h3>\n<p>\u00c0 partir des r\u00e9sultats connus des d\u00e9riv\u00e9es des fonctions exponentielles et logarithmiques, on obtient les r\u00e9sultats fondamentaux suivants :<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int e^{x}dx = e^{x} + C<\/span>\n<br \/>\nParce que <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left(e^x + C\\right) = e^x<\/span>\n<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{1}{x} dx = ln|x| + C<\/span>\n<\/br><br \/>\nParce que <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left(ln|x| + C \\right) = \\dfrac{1}{|x|} sig(x) = \\dfrac{1}{x}<\/span>\n<\/br><br \/>\nO\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">sig(x)<\/span> est la fonction signe d\u00e9finie comme suit :<br \/>\n<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">sig(x) = \\left\\{\\begin{array}{} +1 &amp;,&amp;0\\lt x \\\\ -1 &amp;,&amp; x\\lt 0 \\end{array}\\right.<\/span><\/center>\n<\/li>\n<\/ol>\n<p>Le r\u00e9sultat de l\u2019int\u00e9grale de <span class=\"katex-eq\" data-katex-display=\"false\">1\/x<\/span> nous permet d\u2019\u00e9largir notre capacit\u00e9 \u00e0 int\u00e9grer des fonctions, puisque nous pouvons commencer \u00e0 int\u00e9grer des fonctions qui consistent en un quotient de polyn\u00f4mes.<\/p>\n<div style=\"background-color:#F3FFF3; padding:20px;\">\n<p><strong>Exemple :<\/strong><\/p>\n<ol>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{x^2 + 3x + 2}{5x^2}dx = \\int \\dfrac{1}{5} + \\dfrac{3}{5}\\dfrac{1}{x} + \\dfrac{2}{5}\\dfrac{1}{x^2}dx<\/span>\n<\/br><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\dfrac{x}{5}+\\dfrac{3}{5}ln(x) - \\dfrac{2}{5}\\dfrac{1}{x} + C <\/span><\/li>\n<p><\/br><\/p>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{x^2 - 3 x + 2}{(x-2)^2}dx = \\int \\dfrac{(x-2)^2 + (x-2)}{(x-2)^2} dx<\/span><\/li>\n<p><\/br><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">= \\displaystyle \\int 1 + \\dfrac{1}{x-2} dx\\\\ \\\\\n\n= x + \\displaystyle \\int \\dfrac{1}{x-2}dx = x + ln|x-2| + C<\/span>\n<\/br><br \/>\nParce que<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left( ln|x-2| + C\\right) = \\dfrac{1}{|x-2|}sig(x-2) = \\dfrac{1}{x-2}<\/span>\n<\/ol>\n<\/div>\n<h3>Int\u00e9grales de fonctions hyperboliques de base<\/h3>\n<p>Les fonctions hyperboliques de base sont :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} sinh(x) &amp;=&amp; \\dfrac{e^x - e^{-x}}{2} \\\\ \\\\\n\ncosh(x) &amp;=&amp; \\dfrac{e^x + e^{-x}}{2}\n\n\\end{array}<\/span>\n<p>\u00c9tant donn\u00e9 que nous avons d\u00e9j\u00e0 vu comment fonctionne l\u2019int\u00e9grale de la fonction exponentielle, nous n\u2019aurons aucun probl\u00e8me avec les int\u00e9grales du sinus et du cosinus hyperbolique.<\/p>\n<p>Pour le sinus hyperbolique, le calcul est pratiquement direct :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int sinh(x) dx\n\n&amp;=&amp; \\displaystyle \\int \\dfrac{e^x - e^{-x}}{2}dx \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left( \\displaystyle \\int e^x dx - \\int e^{-x}  dx \\right) \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left(e^x + e^{-x} \\right) + C = cosh(x) + C\n\n\\end{array}<\/span>\n<p>Et pour le cosinus hyperbolique, les calculs sont pratiquement analogues :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int cosh(x) dx\n\n&amp;=&amp; \\displaystyle \\int \\dfrac{e^x + e^{-x}}{2}dx \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left( \\displaystyle \\int e^x dx + \\int e^{-x}  dx \\right) \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left(e^x - e^{-x} \\right) + C = sinh(x) + C\n\n\\end{array}<\/span>\n<p>En plus de celles-ci, il existe de nombreuses autres fonctions hyperboliques que l\u2019on peut int\u00e9grer :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} tanh(x) &amp;=&amp; \\dfrac{sinh(x)}{cosh(x)} \\\\\n\nsech(x) &amp;=&amp; \\dfrac{1}{cosh(x)} \\\\\n\n{}csch(x) &amp;=&amp; \\dfrac{1}{sinh(x)} \\\\\n\nctgh(x) &amp;=&amp; \\dfrac{1}{tanh(x)}\n\n\\end{array}<\/span>\n<p>Cependant, leur int\u00e9gration n\u00e9cessite d\u2019autres techniques que nous verrons dans les prochains cours.<\/p>\n<h3>Int\u00e9grales de fonctions trigonom\u00e9triques de base<\/h3>\n<p>Les fonctions trigonom\u00e9triques de base sont <span class=\"katex-eq\" data-katex-display=\"false\">sin(x)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">cos(x)<\/span>. Le calcul de leurs int\u00e9grales est pratiquement direct \u00e0 partir de ce que nous savons d\u00e9j\u00e0 de leurs d\u00e9riv\u00e9es.<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int sin(x) dx = -cos(x) + C \\\\ \\\\\n\n{} \\displaystyle \\int cos(x) dx = sen(x) + C\n\n\\end{array}<\/span>\n<p>Cela s\u2019explique par le fait que :<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{}  \\dfrac{d}{dx}\\left( sin(x) + C \\right) &amp;=&amp; cos(x) \\\\ \\\\\n\n{}  \\dfrac{d}{dx}\\left( cos(x) + C \\right) &amp;=&amp; -sin(x) \\\\ \\\\\n\n\\end{array}<\/span>\n<h2>Conclusion<\/h2>\n<p>Dans ce cours, nous avons explor\u00e9 les int\u00e9grales ind\u00e9finies depuis leurs fondements th\u00e9oriques jusqu\u2019\u00e0 leurs applications pratiques les plus \u00e9l\u00e9mentaires. Nous avons appris \u00e0 les reconna\u00eetre comme le processus inverse de la d\u00e9rivation, \u00e0 identifier leurs propri\u00e9t\u00e9s de base et \u00e0 appliquer des techniques directes pour int\u00e9grer des fonctions polynomiales, exponentielles, logarithmiques, hyperboliques et trigonom\u00e9triques simples. Ces connaissances constituent une base essentielle pour aborder des probl\u00e8mes d\u2019int\u00e9gration plus complexes \u00e0 l\u2019avenir, et seront fondamentales pour l\u2019\u00e9tude d\u2019applications avanc\u00e9es en physique, ing\u00e9nierie et autres sciences. Avec ces connaissances de base, il sera possible d\u2019introduire des techniques plus sophistiqu\u00e9es dans les cours suivants.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Int\u00e9grales Ind\u00e9finies et Techniques de Base d\u2019Int\u00e9gration Dans ce cours, les techniques de base pour calculer les int\u00e9grales ind\u00e9finies les plus \u00e9l\u00e9mentaires sont introduites, ainsi que les propri\u00e9t\u00e9s de l\u2019op\u00e9rateur d\u2019int\u00e9gration. Cela comprend les int\u00e9grales polynomiales, exponentielles, hyperboliques et trigonom\u00e9triques de base. Objectifs d\u2019Apprentissage :\u00c0 la fin de ce cours, l\u2019\u00e9tudiant sera capable de Comprendre [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32629,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":24,"footnotes":""},"categories":[1145,569],"tags":[],"class_list":["post-32670","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calcul-integral","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Int\u00e9grales Ind\u00e9finies et Techniques de Base d\u2019Int\u00e9gration - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Ma\u00eetrise les techniques de base d\u2019int\u00e9gration avec des exemples clairs et des explications \u00e9tape par \u00e9tape. 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