{"id":11434,"date":"2025-08-18T15:12:21","date_gmt":"2025-08-18T13:12:21","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=11434"},"modified":"2025-08-18T15:12:32","modified_gmt":"2025-08-18T13:12:32","slug":"integrale-et-decomposition-en-elements-simples","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/","title":{"rendered":"Int\u00e9grale et d\u00e9composition en \u00e9l\u00e9ments simples"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Le but est de d\u00e9terminer la valeur de l&rsquo;int\u00e9grale:$$I=\\int_0^1 \\frac{x}{(x^2+5x+6)^2}\\text{d}x$$ \u00e0 l&rsquo;aide d&rsquo;une d\u00e9composition en \u00e9l\u00e9ments simples.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u00c0 premi\u00e8re vue, la fonction $f:x\\mapsto \\frac{x}{(x^2+5x+6)^2}$ n&rsquo;a pas de primitive simple.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons donc d\u00e9composer en \u00e9l\u00e9ments simples cette fonction \u00e0 l&rsquo;aide d&rsquo;une technique rapide.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" 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ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/#Integrale_et_decomposition_en_elements_simples_recherche_des_coefficients\" >Int\u00e9grale et d\u00e9composition en \u00e9l\u00e9ments simples: recherche des coefficients<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/#Recherche_de_%C2%AB_b_%C2%BB\" >Recherche de \u00ab\u00a0b\u00a0\u00bb<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/#Recherche_de_%C2%AB_d_%C2%BB\" >Recherche de \u00ab\u00a0d\u00a0\u00bb<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/#Premier_bilan\" >Premier bilan<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/#Recherche_de_%C2%AB_a_%C2%BB\" >Recherche de \u00ab\u00a0a\u00a0\u00bb<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/#Recherche_de_%C2%AB_c_%C2%BB\" >Recherche de \u00ab\u00a0c\u00a0\u00bb<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/#Decomposition_finale\" >D\u00e9composition finale<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/#Integrale_et_decomposition_en_elements_simples_calcul_de_lintegrale\" >Int\u00e9grale et d\u00e9composition en \u00e9l\u00e9ments simples: calcul de l&rsquo;int\u00e9grale<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Integrale_et_decomposition_en_elements_simples_la_forme_souhaitee\"><\/span>Int\u00e9grale et d\u00e9composition en \u00e9l\u00e9ments simples: la forme souhait\u00e9e<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">D\u00e9composer en \u00e9l\u00e9ments simples une telle fraction revient \u00e0 \u00e9crire la fraction comme somme d&rsquo;autres fractions plus simples.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Commen\u00e7ons donc par observer que le polyn\u00f4me $x^2+5x+6$ admet deux racines r\u00e9elles: -2 et -3. Ainsi, on peut \u00e9crire:$$f(x)=\\frac{x}{(x+2)^2(x+3)^2}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;id\u00e9e consiste donc \u00e0 \u00e9crire $f(x)$ sous la forme:$$f(x)=\\frac{a}{x+2} + \\frac{b}{(x+2)^2} + \\frac{c}{x+3} + \\frac{d}{(x+3)^2}.\\quad(1)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Bien s\u00fbr, on pourrait r\u00e9duire au m\u00eame d\u00e9nominateur l&rsquo;expression \u00e0 droite du signe \u00e9gal, mais c&rsquo;est tout de m\u00eame assez long et fastidieux&#8230; On va donc utiliser une autre m\u00e9thode&#8230;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Integrale_et_decomposition_en_elements_simples_recherche_des_coefficients\"><\/span>Int\u00e9grale et d\u00e9composition en \u00e9l\u00e9ments simples: recherche des coefficients<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Recherche_de_%C2%AB_b_%C2%BB\"><\/span>Recherche de \u00ab\u00a0b\u00a0\u00bb<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Multiplions par $(x+2)^2$ les membres de l&rsquo;\u00e9galit\u00e9 (1). On obtient:$$(x+2)^2f(x)=\\frac{x}{(x+3)^2}=a(x+2) + b + \\frac{c(x+2)^2}{x+3} + \\frac{d(x+2)^2}{(x+3)^2}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Consid\u00e9rons maintenant cette \u00e9galit\u00e9 lorsque $x$ tend vers $-2$; on obtient alors:$$\\frac{-2}{(-2+3)^2}=b$$soit:$$b=-2.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Recherche_de_%C2%AB_d_%C2%BB\"><\/span>Recherche de \u00ab\u00a0d\u00a0\u00bb<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Nous allons faire de m\u00eame pour trouver $d$: nous allons multiplier les membres de l&rsquo;\u00e9galit\u00e9 (1) par $(x+3)^2$:$$(x+3)^2f(x)=\\frac{x}{(x+2)^2}=\\frac{a(x+3)^2}{x+2} + \\frac{b(x+3)^2}{(x+2)^2} + c(x+3) + d.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Maintenant, prenons la limite quand $x$ tend vers $-3$; on obtient:$$-3=d.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Premier_bilan\"><\/span>Premier bilan<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Nous pouvons alors \u00e9crire \u00e0 ce stade:$$f(x)=\\frac{a}{x+2} + \\frac{-2}{(x+2)^2} + \\frac{c}{x+3} + \\frac{-3}{(x+3)^2}$$soit:$$f(x)+ \\frac{2}{(x+2)^2} + \\frac{3}{(x+3)^2}=\\frac{a}{x+2} + \\frac{c}{x+3}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On peut calculer l&rsquo;expression de droite, ce qui donne:$$\\frac{5}{(x+2)(x+3)}=\\frac{a}{x+2} + \\frac{c}{x+3}.\\quad(2)$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Recherche_de_%C2%AB_a_%C2%BB\"><\/span>Recherche de \u00ab\u00a0a\u00a0\u00bb<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Multiplions par $(x+2)$ l&rsquo;\u00e9galit\u00e9 (2); on obtient:$$\\frac{5}{x+3}=a+\\frac{c(x+2)}{x+3}.$$Maintenant, prenons la limite lorsque $x$ tend vers $-2$. On obtient alors:$$5=a.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Recherche_de_%C2%AB_c_%C2%BB\"><\/span>Recherche de \u00ab\u00a0c\u00a0\u00bb<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Multiplions par $(x+3)$ l&rsquo;\u00e9galit\u00e9 (2); on obtient:$$\\frac{5}{x+2}=\\frac{a(x+3)}{x+2}+c.$$Maintenant, prenons la limite lorsque $x$ tend vers $-3$. On obtient alors:$$-5=c.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Decomposition_finale\"><\/span>D\u00e9composition finale<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">On obtient alors:$$f(x)=\\frac{5}{x+2} &#8211; \\frac{2}{(x+2)^2} &#8211; \\frac{5}{x+3} &#8211; \\frac{3}{(x+3)^2}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Integrale_et_decomposition_en_elements_simples_calcul_de_lintegrale\"><\/span>Int\u00e9grale et d\u00e9composition en \u00e9l\u00e9ments simples: calcul de l&rsquo;int\u00e9grale<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">En utilisant la lin\u00e9arit\u00e9 de l&rsquo;int\u00e9grale, on a:$$I=\\int_0^1 \\frac{5}{x+2}\\text{d}x &#8211; \\int_0^1\\frac{2}{(x+2)^2}\\text{d}x &#8211; \\int_0^1\\frac{5}{x+3}\\text{d}x &#8211; \\int_0^1\\frac{3}{(x+3)^2}\\text{d}x,$$quatre int\u00e9grales que l&rsquo;on peut facilement calculer car nous pouvons trouver une primitive des quatre fonctions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On a alors:$$I = \\left[ 5\\ln(x+2) + \\frac{2}{x+2} &#8211; 5\\ln(x+3) + \\frac{3}{x+3} \\right]_0^1$$que l&rsquo;on peut aussi \u00e9crire:$$I = \\left[ 5\\ln\\left(\\frac{x+2}{x+3}\\right) + \\frac{5x + 12}{(x+2)(x+3)} \\right]_0^1.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Finalement,$$I=5\\ln\\frac{3}{4}+\\frac{17}{12}-5\\ln\\frac{2}{3}-\\frac{12}{6} = 5\\ln\\frac{9}{8}-\\frac{7}{12}.$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">La fonction $f$ \u00e9tant positive sur [0;1], cette int\u00e9grale nous donne l&rsquo;aire du domaine d\u00e9limit\u00e9 par la courbe repr\u00e9sentative de $f$ et l&rsquo;axe des abscisses sur ce m\u00eame intervalle:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium has-custom-border\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/08\/integrale.webp\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"79\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/08\/integrale-300x79.webp\" alt=\"int\u00e9grale d\u00e9composition \u00e9l\u00e9ments simples\" class=\"wp-image-11445\" style=\"border-radius:51px\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/08\/integrale-300x79.webp 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/08\/integrale-1024x271.webp 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/08\/integrale-768x203.webp 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2025\/08\/integrale.webp 1200w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Le but est de d\u00e9terminer la valeur de l&rsquo;int\u00e9grale:$$I=\\int_0^1 \\frac{x}{(x^2+5x+6)^2}\\text{d}x$$ \u00e0 l&rsquo;aide d&rsquo;une d\u00e9composition en \u00e9l\u00e9ments simples. \u00c0 premi\u00e8re vue, la fonction $f:x\\mapsto \\frac{x}{(x^2+5x+6)^2}$ n&rsquo;a pas de primitive simple. Nous allons donc d\u00e9composer en \u00e9l\u00e9ments simples cette fonction \u00e0 l&rsquo;aide d&rsquo;une technique rapide.<\/p>\n","protected":false},"author":1,"featured_media":11446,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-11434","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Int\u00e9grale et d\u00e9composition en \u00e9l\u00e9ments simples - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Le but est de d\u00e9terminer la valeur de l&#039;int\u00e9grale d&#039;une fraction rationnelle \u00e0 l&#039;aide d&#039;une d\u00e9composition en \u00e9l\u00e9ments simples.\" \/>\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.mathweb.fr\/euclide\/2025\/08\/18\/integrale-et-decomposition-en-elements-simples\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Int\u00e9grale et d\u00e9composition en \u00e9l\u00e9ments simples - 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