{"id":8599,"date":"2024-01-02T00:00:11","date_gmt":"2024-01-01T18:30:11","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8599"},"modified":"2024-01-17T23:17:57","modified_gmt":"2024-01-17T17:47:57","slug":"integration-of-greatest-integer-function","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-greatest-integer-function\/","title":{"rendered":"Integration of greatest integer function"},"content":{"rendered":"\n<p>The integration of the greatest integer function, often denoted as [x], where x is a real number, can be a bit tricky because the function is not continuous. <br>The greatest integer function returns the largest integer less than or equal to x. For example,<br>[1.2] = 1<br>[-1.1] = -2<br>[4.5]=4<br>[.5]=0<br>[-.2]=-1<br>[2]=2<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Integration of greatest integer function<\/h2>\n\n\n\n<p>To integrate the greatest integer function from a to b, where a and b are real numbers and a &lt; b, you need to consider the discontinuities that occur at integer values. The integral can be expressed as a sum of integrals over intervals where the function is constant.<br>Lets see with an example<br><strong>Example 1<\/strong><br>Evaluate<br>\\[<br>\\int_{1}^{5} [ x ]  \\, dx<br>\\]<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Here both the limits are integers<\/li>\n\n\n\n<li>In this case, the integral can be broken down into the sum of integrals over the intervals [1, 2), [2, 3), [3, 4), and [4, 5). On each of these intervals, the greatest integer function is constant. So, the integral becomes:<\/li>\n<\/ul>\n\n\n\n<p>\\[<br>\\int_{1}^{2} 1 \\, dx + \\int_{2}^{3} 2 \\, dx + \\int_{3}^{4} 3 \\, dx + \\int_{4}^{5} 4 \\, dx<br>\\]<\/p>\n\n\n\n<p>Calculating each integral separately:<\/p>\n\n\n\n<p>\\[<br>= (2 &#8211; 1) \\cdot 1 + (3 &#8211; 2) \\cdot 2 + (4 &#8211; 3) \\cdot 3 + (5 &#8211; 4) \\cdot 4<br>\\]<br>\\[<br>= 1 \\cdot 1 + 1 \\cdot 2 + 1 \\cdot 3 + 1 \\cdot 4<br>\\]<br>\\[<br>= 1 + 2 + 3 + 4 = 10<br>\\]<\/p>\n\n\n\n<p><strong>Example 2<\/strong><br>Evaluate<\/p>\n\n\n\n<p>\\[<br>\\int_{2.5}^{6.5} [x ]  \\, dx<br>\\]<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Here both the limits are non-integers<\/li>\n\n\n\n<li>This integral can be broken down into the sum of integrals over the intervals [2.5, 3), [3, 4), [4, 5), [5, 6), and [6, 6.5). On each of these intervals, the greatest integer function takes on a constant value. So, the integral becomes:<\/li>\n<\/ul>\n\n\n\n<p>\\[<br>\\int_{2.5}^{3} 2 \\, dx + \\int_{3}^{4} 3 \\, dx + \\int_{4}^{5} 4 \\, dx + \\int_{5}^{6} 5 \\, dx + \\int_{6}^{6.5} 6 \\, dx<br>\\]<\/p>\n\n\n\n<p>Calculating each integral separately:<\/p>\n\n\n\n<p>\\[<br>= (3 &#8211; 2.5) \\cdot 2 + (4 &#8211; 3) \\cdot 3 + (5 &#8211; 4) \\cdot 4 + (6 &#8211; 5) \\cdot 5 + (6.5 &#8211; 6) \\cdot 6<br>\\]<br>\\[<br>= 0.5 \\cdot 2 + 1 \\cdot 3 + 1 \\cdot 4 + 1 \\cdot 5 + 0.5 \\cdot 6<br>\\]<br>\\[<br>= 1 + 3 + 4 + 5 + 3 = 16<br>\\]<\/p>\n\n\n\n<p><strong>General Formula<\/strong><\/p>\n\n\n\n<p>Now we have seen some examples, lets see if we can determine some formula based on it<\/p>\n\n\n\n<p>There are four case here<br>(I)<br>Let&#8217;s say a and b are not integers<\/p>\n\n\n\n<p>If ( n ) is an integer such that ( a &lt; n &lt; b ), the function [x] is constant and equal to ( n &#8211; 1 ) for ( x ) in the interval ( [n-1, n) ). Therefore, the integral over this interval is simply the length of the interval times the value of the function.<\/p>\n\n\n\n<p>The integral from a to b of the greatest integer function [x] is:<\/p>\n\n\n\n<p>\\[<br>\\int_{a}^{b} [x] \\, dx = \\int_{a}^{[a+1]} [x] \\, dx + \\sum_{n = [a+1]}^{[b] &#8211; 1} \\int_{n}^{n+1} [x] \\, dx + \\int_{[b]}^{b} [x] \\, dx<br>\\]<\/p>\n\n\n\n<p>In each interval [n, n+1), the value of [x] is just n, so the integral becomes:<\/p>\n\n\n\n<p>\\[<br>= [a] \\cdot ([a+1] &#8211; a) + \\sum_{n = [a+1]}^{[b] &#8211; 1} n + [b] \\cdot (b &#8211; [b])<br>\\]<\/p>\n\n\n\n<p>This formula breaks the integral into parts where the function is constant, and then sums those parts up.<\/p>\n\n\n\n<p>(II) if a and b are integers, the corresponding term simplifies since both ( [a] = a ) and ( [b] = b ).<br>\\[<br>\\int_{a}^{b} [x] \\, dx = \\sum_{n = [a+1]}^{[b] &#8211; 1} n +a= \\sum_{n = [a]}^{[b] &#8211; 1} n<br>\\]<\/p>\n\n\n\n<p>(III)<br>if a is integer and b is not, the corresponding term simplifies since ( [a] = a ) .<\/p>\n\n\n\n<p>\\[<br>\\int_{a}^{b} [x] \\, dx = \\sum_{n = [a]}^{[b] &#8211; 1} n + [b] \\cdot (b &#8211; [b])<br>\\]<\/p>\n\n\n\n<p>(III)<br>if a is not and b is a inetgers, the corresponding term simplifies since ( [b] = b ) .<\/p>\n\n\n\n<p>\\[<br>\\int_{a}^{b} [x] \\, dx = \\sum_{n = [a+1]}^{[b] &#8211; 1} n + [a] \\cdot ([a+1] &#8211; a)<br>\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved Examples<\/h2>\n\n\n\n<p><strong>Question 1<\/strong><br>Find<\/p>\n\n\n\n<p>$\\int_{1}^{2} [3x] \\, dx$<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p>let 3x=t<br>3 dx=dt<br>dx= dt\/3<br>Therefore<\/p>\n\n\n\n<p>$\\int_{1}^{2} [3x] \\, dx=\\frac {1}{3} \\int_{3}^{6} [t] \\, dt$<br>$= \\frac {1}{3} [ \\int_{3}^{4} 3 \\, dt + \\int_{4}^{5} 4 \\, dt +   \\int_{5}^{6} 5 \\, dt] =4$<\/p>\n\n\n\n<p><strong>Question <\/strong>2<br>Find<\/p>\n\n\n\n<p>$\\int_{0}^{2\\pi} [sin x] \\, dx$<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p> $\\int_{0}^{2\\pi} [sin x] \\, dx = \\int_{0}^{\\pi} 0 \\, dx + \\int_{\\pi}^{2\\pi} -1 \\, dx= -\\pi$<\/p>\n\n\n\n<div class=\"wp-block-group has-ast-global-color-4-background-color has-background is-layout-constrained wp-container-core-group-is-layout-ca99af60 wp-block-group-is-layout-constrained\" style=\"border-radius:17px;margin-top:var(--wp--preset--spacing--20);margin-bottom:var(--wp--preset--spacing--20);padding-top:var(--wp--preset--spacing--20);padding-right:var(--wp--preset--spacing--50);padding-bottom:var(--wp--preset--spacing--20);padding-left:var(--wp--preset--spacing--50)\"><div class=\"wp-block-group__inner-container\">\n<p class=\"has-medium-font-size\">Other Integration Related Articles<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\" style=\"padding-right:0;padding-left:0;font-size:16px\"><table style=\"border-style:none;border-width:0px\"><tbody><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-tan-inverse-x\/\">Integration of tan inverse x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-cos-inverse-x\/\">Integration of cos inverse x<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-cos-square-x\/\">Integration of cos square x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-fractional-part-of-x\/\">Integration of fractional part of x<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-uv-formula\/\">integration of uv formula<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-log-x\/\">integration of log x<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-sin-inverse-x\/\">integration of sin inverse x<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-constant\/\">Integration of constant term<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-hyperbolic-functions\/\">integration of hyperbolic functions<\/a><\/td><td><a data-type=\"post\" data-id=\"5701\" href=\"https:\/\/physicscatalyst.com\/article\/integration-formulas-list\/\">List of Integration Formulas <\/a><\/td><\/tr><tr><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-log-sinx\/\">integration of log sinx<\/a><\/td><td><a href=\"https:\/\/physicscatalyst.com\/article\/integration-of-sin-x-cos-x-dx\/\">integration of sin x cos x dx<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>The integration of the greatest integer function, often denoted as [x], where x is a real number, can be a bit tricky because the function is not continuous. The greatest integer function returns the largest integer less than or equal to x. For example,[1.2] = 1[-1.1] = -2[4.5]=4[.5]=0[-.2]=-1[2]=2 Integration of greatest integer function To integrate [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[506],"class_list":["post-8599","post","type-post","status-publish","format-standard","hentry","category-maths","tag-integration"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Integration of greatest integer function - physicscatalyst&#039;s Blog<\/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:\/\/physicscatalyst.com\/article\/integration-of-greatest-integer-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of greatest integer function - physicscatalyst&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"The integration of the greatest integer function, often denoted as [x], where x is a real number, can be a bit tricky because the function is not continuous. 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The greatest integer function returns the largest integer less than or equal to x. For example,[1.2] = 1[-1.1] = -2[4.5]=4[.5]=0[-.2]=-1[2]=2 Integration of greatest integer function To integrate&hellip;","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8599","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=8599"}],"version-history":[{"count":6,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8599\/revisions"}],"predecessor-version":[{"id":8803,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8599\/revisions\/8803"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=8599"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=8599"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=8599"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}