{"id":263,"date":"2023-05-20T20:04:53","date_gmt":"2023-05-20T20:04:53","guid":{"rendered":"http:\/\/matematicastro.es\/?p=263"},"modified":"2023-05-20T20:04:53","modified_gmt":"2023-05-20T20:04:53","slug":"integrales-indefinidas-calculo-de-primitivas-i","status":"publish","type":"post","link":"https:\/\/matematicastro.es\/?p=263","title":{"rendered":"Integrales indefinidas. C\u00e1lculo de primitivas (I)"},"content":{"rendered":"\n<p>Utilizando distintos <a href=\"https:\/\/www.dropbox.com\/s\/3cnr5t2lkqnt2xq\/metodos_integracion.pdf?dl=0\">m\u00e9todos de integraci\u00f3n<\/a> se resuelven muchas integrales al nivel de 2\u00ba de Bachillerato Cient\u00edfico-T\u00e9cnico (en la materia de Matem\u00e1ticas II).<\/p>\n\n\n\n<p>$$\\int{\\cos^2x\\,dx=\\begin{bmatrix}u=\\cos x&amp;\\text{;}&amp;du=-\\text{sen}\\,x\\,dx\\\\dv=\\cos x\\,dx&amp;\\text{;}&amp;v=\\text{sen}\\,x\\end{bmatrix}}=$$<\/p>\n\n\n\n<p>$$=\\text{sen}\\,x\\cos x+\\int{\\text{sen}^2x\\,dx}=\\text{sen}\\,x\\cos x+x-\\int{\\cos^2x\\,dx}\\Rightarrow$$<\/p>\n\n\n\n<p>$$2\\int{\\cos^2x\\,dx}=x+\\text{sen}\\,x\\cos x\\Rightarrow\\int{\\cos^2x\\,dx}=\\frac{x+\\text{sen}\\,x\\cos x}{2}+C$$<\/p>\n\n\n\n<p>Hay otra forma m\u00e1s r\u00e1pida de hacer esta integral, pero hemos de recordar una f\u00f3rmula trigonom\u00e9trica:<\/p>\n\n\n\n<p>$$\\cos 2x=\\cos^2x-\\text{sen}^2x\\Rightarrow\\cos 2x=\\cos^2x-(1-\\cos^2x)\\Rightarrow$$<\/p>\n\n\n\n<p>$$\\Rightarrow\\cos2x=2\\cos^2x-1\\Rightarrow\\cos^2x=\\frac{\\cos2x+1}{2}$$<\/p>\n\n\n\n<p>Entonces:<\/p>\n\n\n\n<p>$$\\int{\\cos^2x\\,dx}=\\int{\\frac{\\cos2x+1}{2}\\,dx}=\\int{\\frac{1}{2}\\,dx}+\\int{\\frac{\\cos2x}{2}\\,dx}\\Rightarrow$$<\/p>\n\n\n\n<p>$$\\Rightarrow\\int{\\cos^2x\\,dx}=\\frac{x}{2}+\\frac{\\text{sen}\\,2x}{4}+C=\\frac{x+\\text{sen}\\,x\\cos x}{2}+C$$<\/p>\n\n\n\n<p>Utilizando lo anterior:<\/p>\n\n\n\n<p>$$\\int{\\text{sen}^2x\\,dx}=\\int{(1-\\cos^2x)\\,dx}=\\int{1\\,dx}-\\int{\\cos^2x\\,dx}\\Rightarrow$$<\/p>\n\n\n\n<p>$$\\int{\\text{sen}^2x\\,dx}=x-\\frac{x+\\text{sen}\\,x\\cos x}{2}+C=\\frac{x-\\text{sen}\\,x\\cos x}{2}+C$$<\/p>\n\n\n\n<p>Otra integral f\u00e1cil de hacer por partes es la siguiente:<\/p>\n\n\n\n<p>$$\\int{x\\cos x\\,dx}=\\begin{bmatrix}u=x &amp; \\text{;} &amp; du=dx\\\\dv=\\cos x\\,dx&amp;\\text{;}&amp;v=\\text{sen}\\,x\\end{bmatrix}=$$<\/p>\n\n\n\n<p>$$=x\\,\\text{sen}\\,x-\\int{\\text{sen}\\,x\\,dx}\\Rightarrow\\int{x\\cos x\\,dx}=x\\,\\text{sen}\\,x+\\cos x+C$$<\/p>\n\n\n\n<p>De manera completamente an\u00e1loga a la anterior:<\/p>\n\n\n\n<p>$$\\int{x\\,\\text{sen}\\,x\\,dx}=-x\\cos x+\\text{sen}\\,x+C$$<\/p>\n\n\n\n<p>Y, de momento, la \u00faltima. Esta es inmediata:<\/p>\n\n\n\n<p>$$\\int{\\text{sen}\\,x\\cos x\\,dx}=\\frac{\\text{sen}^2x}{2}+C$$<\/p>\n\n\n\n<p>Seguiremos en Integrales indefinidas. C\u00e1lculo de primitivas (II).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Utilizando distintos m\u00e9todos de integraci\u00f3n se resuelven muchas integrales al nivel de 2\u00ba de Bachillerato Cient\u00edfico-T\u00e9cnico (en la materia de Matem\u00e1ticas II). $$\\int{\\cos^2x\\,dx=\\begin{bmatrix}u=\\cos x&amp;\\text{;}&amp;du=-\\text{sen}\\,x\\,dx\\\\dv=\\cos x\\,dx&amp;\\text{;}&amp;v=\\text{sen}\\,x\\end{bmatrix}}=$$ $$=\\text{sen}\\,x\\cos x+\\int{\\text{sen}^2x\\,dx}=\\text{sen}\\,x\\cos x+x-\\int{\\cos^2x\\,dx}\\Rightarrow$$ $$2\\int{\\cos^2x\\,dx}=x+\\text{sen}\\,x\\cos x\\Rightarrow\\int{\\cos^2x\\,dx}=\\frac{x+\\text{sen}\\,x\\cos x}{2}+C$$ Hay otra forma m\u00e1s r\u00e1pida de hacer esta integral, pero hemos de recordar una f\u00f3rmula trigonom\u00e9trica: $$\\cos 2x=\\cos^2x-\\text{sen}^2x\\Rightarrow\\cos 2x=\\cos^2x-(1-\\cos^2x)\\Rightarrow$$ $$\\Rightarrow\\cos2x=2\\cos^2x-1\\Rightarrow\\cos^2x=\\frac{\\cos2x+1}{2}$$ Entonces: $$\\int{\\cos^2x\\,dx}=\\int{\\frac{\\cos2x+1}{2}\\,dx}=\\int{\\frac{1}{2}\\,dx}+\\int{\\frac{\\cos2x}{2}\\,dx}\\Rightarrow$$ $$\\Rightarrow\\int{\\cos^2x\\,dx}=\\frac{x}{2}+\\frac{\\text{sen}\\,2x}{4}+C=\\frac{x+\\text{sen}\\,x\\cos x}{2}+C$$ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":277,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,4],"tags":[],"class_list":{"0":"post-263","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","6":"hentry","7":"category-analisis-matematico","8":"category-bachillerato","10":"post-with-thumbnail","11":"post-with-thumbnail-large"},"_links":{"self":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/263","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=263"}],"version-history":[{"count":13,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/263\/revisions"}],"predecessor-version":[{"id":276,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/263\/revisions\/276"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/media\/277"}],"wp:attachment":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=263"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=263"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}