{"id":33424,"date":"2023-04-04T13:00:37","date_gmt":"2023-04-04T13:00:37","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33424"},"modified":"2025-07-23T10:25:20","modified_gmt":"2025-07-23T10:25:20","slug":"integralia-indefinita-et-technicae-fundamentales-integrationis","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/integralia-indefinita-et-technicae-fundamentales-integrationis\/","title":{"rendered":"Integralia Indefinita et Technicae Fundamentales Integrationis"},"content":{"rendered":"<style>\np, ul, ol {\n    text-align: justify;\n}\nh1, h2, h3 {\ntext-align:center;\n}\n<\/style>\n<p><center><\/p>\n<h1>Integraliae Indefinitae et Technicae Fundamentales Integrationis<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\">In hac lectione introducuntur technicae fundamentales ad integralia indefinita simplicissima computanda, itemque proprietates operatoris integrationis. Hoc comprehendit integralia polynomalia, exponentalia, hyperbolica atque trigonometra elementaria.<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>Propositi Discendi<\/u>:<\/strong><br \/>Peracta hac lectione discipulus poterit<\/p>\n<ol>\n<li><strong>Intellegere<\/strong> processum integrationis indefinitae ut processum inversum derivationis.<\/li>\n<li><strong>Computare<\/strong> integralia polynommiorum et expressionum functiones exponentiales, hyperbolicas et trigonometricas continentium.<\/li>\n<li><strong>Adhibere<\/strong> proprietates integraliarum ad manipulationes algebraicas faciendas quae calculum earum faciliorem reddant.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX RERUM<\/strong><br \/>\n<a href=\"#1\">DE MOMENTO INTEGRALIARUM INDEFINITARUM<\/a><br \/>\n<a href=\"#2\">ANTIDERIVATAE, INTEGRALIA INDEFINITA ET FUNCTIONUM PRIMITIVAE<\/a><br \/>\n<a href=\"#3\">TECHNICAE FUNDAMENTALES INTEGRATIONIS<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/4wSTxA7zY9k\" title=\"YouTube video player\" 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>De momento integraliarum indefinitarum<\/h2>\n<p>Integralia indefinita sunt instrumentum fundamentale in calculo, latissimo usu praeditum in scientiis physicis et mathematicis. Permittunt primitivam functionis datae computare, quae vicissim adhibetur ad areas sub curvis, volumina solidorum, probabilitatum calculum et alias multas applicationes in physica, ingenieria, statistica atque oeconomia computandas. Praeterea, integralia indefinita sunt essentialia ad aequationes differentiales solvendas, unde fiunt instrumenta necessaria in plurimis scientiae technologiaeque campis.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/56fMLiVPwDI\" title=\"YouTube video player\" 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>Antiderivatae, integralia indefinita et functionum primitivae<\/h2>\n<p>Si functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span><\/span> habet derivatam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> in quodam intervallo <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> dato, dicitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span><\/span> esse primitiva <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> in illo intervallo.<\/p>\n<p>Notandum est, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span><\/span> est primitiva <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span>, tunc etiam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x) + C<\/span><\/span>, ubi <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est quaelibet constantia realis, est primitiva eiusdem functionis. Hoc scribitur ita:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int f(x) dx = F(x) + C<\/span>\n<p>Constans <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est quod vocatur <strong>constans integrationis<\/strong>, eiusque praesentia indicat primitivam functionis non esse unicam functionem, sed familiam functionum: collectionem omnium functionum quarum derivata est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> in intervallo <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>.<\/p>\n<p>Vocabula antiderivata, primitiva et integrale indefinitum sunt tres modi exprimendi eandem notionem, quibus indiscrete utimur. Summatim, integratio indefinita est processus inversus computationi derivatorum, atque ex hac idea proprietates eius fundamentales oriuntur.<\/p>\n<h3>Proprietates fundamentales integralium indefinitarum<\/h3>\n<p>Ad integralia indefinita computanda, oportet prius cognoscere quasdam proprietates fundamentales, quae directe a proprietatibus derivatorum oriuntur.<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int  \\dfrac{df(x)}{dx} dx = f(x) + C<\/span><\/span><\/br>Quia integratio indefinita est processus inversus derivationis.<\/li>\n<p><\/br><\/p>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\lambda f(x) dx = \\lambda \\int f(x) dx<\/span><\/span><\/br>Ubi <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda<\/span> est quaelibet constantia realis. Hoc fit quia<\/br>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\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>\n<p>Et postea, posito <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{d\\phi(x)}{dx}<\/span>, habemus<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\lambda f(x) dx = \\lambda \\int f(x)dx<\/span>\n<\/li>\n<p><\/br><\/p>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int f(x) + g(x) dx = \\int f(x) dx + \\int g(x) dx <\/span><\/span>\n<p>Hoc similiter demonstrari potest. Consideremus duas functiones <span class=\"katex-eq\" data-katex-display=\"false\">\\phi(x)<\/span> et  <span class=\"katex-eq\" data-katex-display=\"false\">\\psi(x)<\/span> tales ut<\/p>\n<p style=\"text-align: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>\n<p>Ergo habetur<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\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>\n<\/li>\n<\/ol>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Technicae Fundamentales Integrationis<\/h2>\n<p>Exstant technicae fundamentales integrationis quae sinunt nos quasdam integrales indefinitas computare ex effectibus derivationis obtentis. Per has technicas, assequi possumus haec utilia pro integratione eventa:<\/p>\n<h3>Integralia Functionum Polynomialium<\/h3>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int 1 dx = x + C<\/span><\/span>\n<p dir=\"ltr\">Quia  <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} (x + C)= 1 <\/span>\n<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int x^q dx = \\dfrac{x^{q+1}}{q+1}  + C,<\/span> dummodo <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq -1<\/span><\/span>\n<p>Quia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left(\\dfrac{x^{q+1}}{q+1}  + C\\right) = x^q.<\/span><\/span><\/p>\n<\/li>\n<\/ol>\n<p>His effectibus et proprietatibus fundamentalibus adhibitis, sine ullo negotio possumus integrale cuiuslibet polynomii computare.<\/p>\n<div style=\"background-color:#F3FFF3; padding:20px;\">\n<p><strong>Exemplum:<\/strong><\/p>\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 3x+2 \\right) dx =  \\dfrac{3}{2}x^2 + 2x + C<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><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><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 4x^{12} - 7x^{-1\/3} + 1 \\right) dx  <\/span><\/span> <\/li>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>Integralia Exponentialium et Logarithmicarum<\/h3>\n<p>Ex notis effectibus derivatorum functionum exponentialium et logarithmicarum haec fundamentalia eventa obtinentur:<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int e^{x}dx = e^{x} + C<\/span><\/span>\n<p>Quia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left(e^x + C\\right) = e^x<\/span><\/span><\/p>\n<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{1}{x} dx = ln|x| + C<\/span><\/span>\n<p>Quia <span dir=\"ltr\"><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><\/span><\/p>\n<p>Ubi <span class=\"katex-eq\" data-katex-display=\"false\">sig(x)<\/span> est functio signi hoc modo definita:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>\n<\/li>\n<\/ol>\n<p>Exitus integralium <span class=\"katex-eq\" data-katex-display=\"false\">1\/x<\/span> facultatem nostram ad functiones integrandas amplificat, cum iam incipere possimus integrari functiones quae ex ratione polynomiorum constant.<\/p>\n<div style=\"background-color:#F3FFF3; padding:20px;\">\n<p><strong>Exemplum:<\/strong><\/p>\n<ol>\n<li type=\"a\"><\/br>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\displaystyle \\int \\dfrac{x^2 + 3x + 2}{5x^2}dx &amp;= \\displaystyle \\int \\dfrac{1}{5} + \\dfrac{3}{5}\\cdot \\dfrac{1}{x} + \\dfrac{2}{5}\\cdot\\dfrac{1}{x^2}dx \\\\ \\\\\n\n&amp;=\\dfrac{x}{5}+\\dfrac{3}{5}ln(x) - \\dfrac{2}{5}\\dfrac{1}{x} + C\n\n\\end{array}<\/span>\n<\/li>\n<p><\/br><\/p>\n<li type=\"a\"><\/br>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\displaystyle \\int \\dfrac{x^2 - 3 x + 2}{(x-2)^2}dx &amp;= \\displaystyle \\int \\dfrac{(x-2)^2 + (x-2)}{(x-2)^2} dx \\\\ \\\\\n\n&amp;= \\displaystyle \\int 1 + \\dfrac{1}{x-2} dx \\\\ \\\\\n\n&amp;= x + \\displaystyle \\int \\dfrac{1}{x-2}dx = x + ln|x-2| + C\n\n\\end{array}<\/span>\n<p>Quia<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>Integralia Functionum Hyperbolicarum Fundamentalium<\/h3>\n<p>Functiones hyperbolicae fundamentales sunt<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>Cum iam viderimus quomodo integratio functionis exponentialium fiat, nullam difficultatem habebimus in integrandis sinibus et cosinibus hyperbolicis.<\/p>\n<p>Pro sinu hyperbolico calculus est fere directus:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\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 pro cosino hyperbolico, rationes sunt fere analogae:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>Praeter has, multae aliae functiones hyperbolicae exstant quae integrari possunt:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>Tamen earum integratio requirit alias technicas, quas in lectionibus futuris tractabimus.<\/p>\n<h3>Integralia Functionum Trigonometricarum Fundamentalium<\/h3>\n<p>Functiones trigonometricae fundamentales sunt <span class=\"katex-eq\" data-katex-display=\"false\">sin(x)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">cos(x)<\/span>. Calculus earum integralium est fere directus ex notitia derivatorum earundem.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>Hoc fit quia<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><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>Conclusio<\/h2>\n<p>In hac lectione integralia indefinita ab ipsis fundamentis theoreticis usque ad simplicissimas applicationes practicas exploravimus. Didicimus eas esse processum inversum derivationis, proprietates earum fundamentales agnovimus, atque technicis directis adhibitis functiones polynomiales, exponentiales, logarithmicas, hyperbolicas et trigonom\u00e9tricas simplices integravimus. Hae notitiae constituunt fundamentum necessarium ad problemata integrationis difficiliora in futurum tractanda, atque erunt praecipuae in studiis applicationum provectorum in physica, ingenieria et aliis scientiis. Hoc fundamento percepto, possumus technicas subtiliores in sequentibus lectionibus introducere.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Integraliae Indefinitae et Technicae Fundamentales Integrationis In hac lectione introducuntur technicae fundamentales ad integralia indefinita simplicissima computanda, itemque proprietates operatoris integrationis. Hoc comprehendit integralia polynomalia, exponentalia, hyperbolica atque trigonometra elementaria. Propositi Discendi:Peracta hac lectione discipulus poterit Intellegere processum integrationis indefinitae ut processum inversum derivationis. Computare integralia polynommiorum et expressionum functiones exponentiales, hyperbolicas et trigonometricas continentium. [&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":4,"footnotes":""},"categories":[1334,1298],"tags":[],"class_list":["post-33424","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculus-integralis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Integralia Indefinita et Technicae Fundamentales Integrationis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Technicas fundamentales integrationis per exempla clara et expositiones gradatim domina. 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