{"id":324,"date":"2021-09-26T14:43:51","date_gmt":"2021-09-26T12:43:51","guid":{"rendered":"https:\/\/blog.heuresolaire.com\/?p=324"},"modified":"2021-10-28T22:09:48","modified_gmt":"2021-10-28T20:09:48","slug":"integrale-sympathique","status":"publish","type":"post","link":"https:\/\/blog.epicycle.fr\/?p=324","title":{"rendered":"Comment cr\u00e9er une int\u00e9grale nulle \u00e0 partir de n’importe quelle fonction"},"content":{"rendered":"\n

<\/p>\n\n\n\n

Pour tout \"k\in\mathbb{R}^*_+\" et toute fonction \"f\" continue sur \"R_+^*\", on a :<\/p>\n\n\n\n

(1) <\/span>   <\/span>\"\begin{equation*}\int_{\frac{1}{k}}^{k} f\left(x+\frac{1}{x}\right)\frac{\ln(x)}{x}\,dx = 0}\end{equation*}\"<\/p><\/p>\n\n\n\n

Id\u00e9e de preuve :<\/em><\/strong> distinguer l’int\u00e9grale de \"1\" \u00e0 \"k\" et l’int\u00e9grale de \"1/k\" \u00e0 \"1\". On peut alors montrer que cette derni\u00e8re est l’oppos\u00e9 de la premi\u00e8re via le changement de variable \"y = 1/x\" (ou bien on peut montrer que cette int\u00e9grale est son propre oppos\u00e9 via le m\u00eame changement de variable).<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Pour tout et toute fonction continue sur , on a : (1)   Id\u00e9e de preuve : distinguer l’int\u00e9grale de \u00e0 et l’int\u00e9grale de \u00e0 . On peut alors montrer que cette derni\u00e8re est l’oppos\u00e9 de la premi\u00e8re via le changement de variable (ou bien on peut montrer que cette int\u00e9grale est son propre oppos\u00e9 … Continuer la lecture de « Comment cr\u00e9er une int\u00e9grale nulle \u00e0 partir de n’importe quelle fonction »<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-324","post","type-post","status-publish","format-standard","hentry","category-non-classe"],"_links":{"self":[{"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/posts\/324","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=324"}],"version-history":[{"count":20,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/posts\/324\/revisions"}],"predecessor-version":[{"id":374,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=\/wp\/v2\/posts\/324\/revisions\/374"}],"wp:attachment":[{"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=324"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=324"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.epicycle.fr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=324"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}