{"id":5,"date":"2024-03-31T13:16:55","date_gmt":"2024-03-31T12:16:55","guid":{"rendered":"https:\/\/tutors4you.com\/?page_id=5"},"modified":"2024-03-31T13:16:56","modified_gmt":"2024-03-31T12:16:56","slug":"indefinite-integral","status":"publish","type":"page","link":"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/","title":{"rendered":"Indefinite Integral"},"content":{"rendered":"\n
\n\t\t\n\t\t

If ‘f’ and ‘g’ are functions of ‘x’, such\n that g'(x)=f(x) then the function ‘g’ is called an\n integral of ‘f’ with respect to ‘x’, and is written\n symbolically as:<\/p>\n\t\t

f(x)dx = g(x) + c<\/p>\n\t\t

where: f(x) is called the integrand and ‘c’\n is called the constant of integration<\/p>\n\t\t

Note: If d\/dx f(x) = g(x) then\n d\/dx {f(x) + c} = g(x)<\/p>\n\t\t

Where ‘c’ is constant, because\n differentiation of a constant is zero.<\/p>\n\t\t

Thus the general value g(x)dx is\n f(x)+c, where ‘c’ is the constant of integration.<\/p>\n\t\t

Clearly integral will change if ‘c’ changes.\n Thus integral of a function is not unique, hence it is\n called indefinite integral.<\/p>\n\t\t

Standard Results:<\/h3>\n\t\t

These standard results for integral calculus\n are derived directly from the standard results of\n differential calculus<\/p>\n\t\t\n \n\t\t\t\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
Differential\n Calculus<\/th>\n Integral\n Calculus<\/th>\n <\/tr>\n\t\t\t<\/thead>\n\t\t\t
d\/dx(xn+1<\/sup>\/\n n+1) = xn<\/sup><\/td>\n xn\n <\/sup>dx =(xn+1<\/sup>\/n+1) + C [n not\n =1]<\/td>\n <\/tr>\n
d\/dx loge<\/sub>|x|\n = 1\/x<\/td>\n 1\/x dx\n = loge<\/sub>|x| + c [n= -1]<\/td>\n <\/tr>\n
d\/dx ex<\/sup>\n = ex<\/sup><\/td>\n ex\n <\/sup>dx = ex<\/sup> + c<\/td>\n <\/tr>\n
d\/dx ax<\/sup>\n = ax<\/sup> loge<\/sub>a<\/td>\n ax<\/sup>\n dx = ax<\/sup> \/ loge<\/sub>a + c\n [a>0]<\/td>\n <\/tr>\n
d\/dx Cosx =\n – Sinx<\/td>\n Sinx dx\n = – Cosx +c<\/td>\n <\/tr>\n
d\/dx Sinx =\n Cosx<\/td>\n Cosx dx\n = Sinx + c<\/td>\n <\/tr>\n
d\/dx Tanx =\n Sec2<\/sup>x<\/td>\n Sec2<\/sup>x\n dx = Tanx + c <\/td>\n <\/tr>\n
d\/dx Cotx =\n – Cosec2<\/sup> x<\/td>\n Cosec2<\/sup>x\n dx = – Cotx + C<\/td>\n <\/tr>\n
d\/dx Secx =\n Secx.Tanx<\/td>\n Secx.Tanx\n dx = Secx + c<\/td>\n <\/tr>\n
d\/dx Cosecx\n = – Cosecx.Cotx<\/td>\n Cosec.Cotx\n dx = – Cosecx + c<\/td>\n <\/tr>\n
d\/dx Sin-1<\/sup>x\n = 1\/v(1-x2<\/sup>)<\/td>\n 1\/(1-x2<\/sup>)\n dx = Sin-1<\/sup> + c<\/td>\n <\/tr>\n
d\/dx Tan-1<\/sup>x\n = 1\/(1+x2<\/sup>)<\/td>\n 1\/(1+x2<\/sup>)\n dx = Tan-1<\/sup>x + c<\/td>\n <\/tr>\n
d\/dx Sec-1<\/sup>x\n = 1\/xv(x2<\/sup> – 1)<\/td>\n 1\/(x2<\/sup>\n – 1) dx = Sec-1<\/sup> x + C<\/td>\n <\/tr>\n
d\/dx Sin-1<\/sup>x\/a\n = 1\/v(a2<\/sup> + x2<\/sup>)<\/td>\n dx\/v(a2<\/sup>\n – x2<\/sup>) = Sin-1<\/sup>x\/a + c<\/td>\n <\/tr>\n
d\/dx (1\/a)\n Tan-1<\/sup>x\/a = 1\/(x2<\/sup>+a2<\/sup>)<\/td>\n dx\/(x2<\/sup>+a2<\/sup>)\n = 1\/a Tan-1<\/sup>(x\/a) +c<\/td>\n <\/tr>\n
d\/dx (1\/a\n Sec-1<\/sup>x\/a) =1\/xv(x2<\/sup> – a2<\/sup>)<\/td>\n dx\/xv(x2<\/sup>-a2<\/sup>)\n = 1\/a Sec-1<\/sup>x\/a +c<\/td>\n <\/tr>\n
d\/dx Coshx\n = Sinhx<\/td>\n Sinh dx\n = Coshx + c<\/td>\n <\/tr>\n
d\/dx Sinhx\n = Coshx<\/td>\n Coshx\n dx = Sinhx + c<\/td>\n <\/tr>\n
d\/dx Tanhx\n = Sech2<\/sup>x<\/td>\n Sec2<\/sup>x\n dx = Tanhx + c<\/td>\n <\/tr>\n
d\/dx Cothx\n = – Cosech2<\/sup>x<\/td>\n Cosech2<\/sup>x\n dx = – Cothx +c<\/td>\n <\/tr>\n
d\/dx Sechx\n = – Sechx.Tanhx<\/td>\n Sechx.Tanhx\n dx = – Sechx + c<\/td>\n <\/tr>\n
d\/dx\n Cosechx = – Cosechx.Cothx<\/td>\n Cosechx.Cothx\n dx= -Cosechx+c<\/td>\n <\/tr>\n\t\t\t<\/tbody>\n <\/table>\n\t\t<\/div>\n\t<\/div>\n","protected":false},"excerpt":{"rendered":"

If ‘f’ and ‘g’ are functions of ‘x’, such that g'(x)=f(x) then the function ‘g’ is called an integral of ‘f’ with respect to ‘x’, and is written symbolically as: f(x)dx = g(x) + c where: f(x) is called the integrand and ‘c’ is called the constant of integration Note: If d\/dx f(x) = g(x) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-5","page","type-page","status-publish","hentry"],"blocksy_meta":[],"yoast_head":"\nIndefinite Integral - Tutors 4 You<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Indefinite Integral - Tutors 4 You\" \/>\n<meta property=\"og:description\" content=\"If ‘f’ and ‘g’ are functions of ‘x’, such that g'(x)=f(x) then the function ‘g’ is called an integral of ‘f’ with respect to ‘x’, and is written symbolically as: f(x)dx = g(x) + c where: f(x) is called the integrand and ‘c’ is called the constant of integration Note: If d\/dx f(x) = g(x) […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/\" \/>\n<meta property=\"og:site_name\" content=\"Tutors 4 You\" \/>\n<meta property=\"article:modified_time\" content=\"2024-03-31T12:16:56+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/\",\"url\":\"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/\",\"name\":\"Indefinite Integral - Tutors 4 You\",\"isPartOf\":{\"@id\":\"https:\/\/tutors4you.com\/#website\"},\"datePublished\":\"2024-03-31T12:16:55+00:00\",\"dateModified\":\"2024-03-31T12:16:56+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/tutors4you.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Indefinite Integral\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/tutors4you.com\/#website\",\"url\":\"https:\/\/tutors4you.com\/\",\"name\":\"Tutors 4 You\",\"description\":\"Helping you learn\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/tutors4you.com\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Indefinite Integral - Tutors 4 You","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/","og_locale":"en_US","og_type":"article","og_title":"Indefinite Integral - Tutors 4 You","og_description":"If ‘f’ and ‘g’ are functions of ‘x’, such that g'(x)=f(x) then the function ‘g’ is called an integral of ‘f’ with respect to ‘x’, and is written symbolically as: f(x)dx = g(x) + c where: f(x) is called the integrand and ‘c’ is called the constant of integration Note: If d\/dx f(x) = g(x) […]","og_url":"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/","og_site_name":"Tutors 4 You","article_modified_time":"2024-03-31T12:16:56+00:00","twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/","url":"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/","name":"Indefinite Integral - Tutors 4 You","isPartOf":{"@id":"https:\/\/tutors4you.com\/#website"},"datePublished":"2024-03-31T12:16:55+00:00","dateModified":"2024-03-31T12:16:56+00:00","breadcrumb":{"@id":"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/tutors4you.com\/index.php\/indefinite-integral\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/tutors4you.com\/index.php\/indefinite-integral\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/tutors4you.com\/"},{"@type":"ListItem","position":2,"name":"Indefinite Integral"}]},{"@type":"WebSite","@id":"https:\/\/tutors4you.com\/#website","url":"https:\/\/tutors4you.com\/","name":"Tutors 4 You","description":"Helping you learn","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/tutors4you.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"}]}},"_links":{"self":[{"href":"https:\/\/tutors4you.com\/index.php\/wp-json\/wp\/v2\/pages\/5","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tutors4you.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/tutors4you.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/tutors4you.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tutors4you.com\/index.php\/wp-json\/wp\/v2\/comments?post=5"}],"version-history":[{"count":4,"href":"https:\/\/tutors4you.com\/index.php\/wp-json\/wp\/v2\/pages\/5\/revisions"}],"predecessor-version":[{"id":12,"href":"https:\/\/tutors4you.com\/index.php\/wp-json\/wp\/v2\/pages\/5\/revisions\/12"}],"wp:attachment":[{"href":"https:\/\/tutors4you.com\/index.php\/wp-json\/wp\/v2\/media?parent=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}