{"id":37955,"date":"2026-02-12T11:57:16","date_gmt":"2026-02-12T11:57:16","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=37955"},"modified":"2026-03-12T07:50:49","modified_gmt":"2026-03-12T07:50:49","slug":"queens-property-in-definite-integrals","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/queens-property-in-definite-integrals\/","title":{"rendered":"How to Use the Queen\u2019s Property in Integrals"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"37955\" class=\"elementor elementor-37955\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<div class=\"elementor-inner\">\n\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-5b87d1a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5b87d1a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5fd5461\" data-id=\"5fd5461\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-53b2ef5 elementor-widget elementor-widget-text-editor\" data-id=\"53b2ef5\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"color: #000000;\" data-sheets-root=\"1\"><strong>Update:<\/strong> This article was last updated on 12th March 2026 to reflect the accuracy and up-to-date information on the page.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-634fe2e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"634fe2e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-58202ff\" data-id=\"58202ff\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-287946b elementor-widget elementor-widget-text-editor\" data-id=\"287946b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400; color: #000000;\">You&#8217;re staring at a definite integral packed with sines, cosines, and fractions. It looks impossible. Your first instinct? Reach for a calculator. But before you do, what if there was a two-step royal shortcut that dissolves the entire problem in under a minute?<\/span><\/p><p><span style=\"font-weight: 400; color: #000000;\">In the world of calculus, two powerful properties, King&#8217;s rule integration and Queen&#8217;s rule integration, work as a dynamic duo. Used together, they can turn a nightmare integral into a clean, simple answer. This guide will walk you through both when to use each and exactly why the Queen steps in when the King falls short.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-5ac510b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5ac510b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1d3a1ad\" data-id=\"1d3a1ad\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9b1452b elementor-widget elementor-widget-text-editor\" data-id=\"9b1452b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #000000;\"><b>What Is King&#8217;s Rule in Integration?<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-aa4b04e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"aa4b04e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-08db33e\" data-id=\"08db33e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6f02813 elementor-widget elementor-widget-text-editor\" data-id=\"6f02813\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">King&#8217;s rule integration (also called the King&#8217;s property of definite integrals) is one of the most elegant symmetry tricks in calculus. It states that a definite integral from <\/span><i><span style=\"font-weight: 400;\">a<\/span><\/i><span style=\"font-weight: 400;\"> to <\/span><i><span style=\"font-weight: 400;\">b<\/span><\/i><span style=\"font-weight: 400;\"> remains unchanged when you substitute <\/span><i><span style=\"font-weight: 400;\">x<\/span><\/i><span style=\"font-weight: 400;\"> with <\/span><i><span style=\"font-weight: 400;\">(a + b \u2212 x)<\/span><\/i><span style=\"font-weight: 400;\">.<\/span><\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-97750fa elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"97750fa\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-577dde1\" data-id=\"577dde1\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6f8f0e0 elementor-widget elementor-widget-text-editor\" data-id=\"6f8f0e0\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<table><tbody><tr><td><h4 style=\"text-align: center;\"><span style=\"color: #ff6600;\"><b>King&#8217;s Rule Formula: <\/b><span style=\"font-weight: 400;\">\u222b[a to b] f(x) dx\u00a0 =\u00a0 \u222b[a to b] f(a + b \u2212 x) dx<\/span><\/span><\/h4><\/td><\/tr><\/tbody><\/table>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-31000a9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"31000a9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-519469f\" data-id=\"519469f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ebd8f6e elementor-widget elementor-widget-text-editor\" data-id=\"ebd8f6e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400; color: #000000;\">This substitution essentially mirrors the function about the midpoint of the interval without changing the integral&#8217;s value. It&#8217;s especially powerful for integrals involving trigonometric functions over symmetric intervals like [0, \u03c0].<\/span><\/p><h4><span style=\"color: #800080;\"><b>When Does King&#8217;s Rule Work Best?<\/b><\/span><\/h4><p><span style=\"font-weight: 400; color: #000000;\">King&#8217;s rule integration is your go-to tool when:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">The integrand involves sin(x), cos(x), or tan(x) over [0, \u03c0] or [0, \u03c0\/2]<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Substituting (a + b \u2212 x) simplifies or cancels part of the function<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">The resulting expression can be added to the original to give a constant integrand<\/span><\/li><\/ul>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-99a041e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"99a041e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-eb7eacb\" data-id=\"eb7eacb\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d95b197 elementor-widget elementor-widget-text-editor\" data-id=\"d95b197\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #000000;\"><b>When Does King&#8217;s Rule Fail?<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-50e20e6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"50e20e6\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-da27b5a\" data-id=\"da27b5a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-25b1673 elementor-widget elementor-widget-text-editor\" data-id=\"25b1673\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Here&#8217;s where students get stuck. Sometimes, King&#8217;s rule integration loops back on itself. You substitute <\/span><i><span style=\"font-weight: 400;\">(a + b \u2212 x)<\/span><\/i><span style=\"font-weight: 400;\"> and end up proving only that <\/span><i><span style=\"font-weight: 400;\">I = I<\/span><\/i><span style=\"font-weight: 400;\"> \u2014, which is mathematically true but completely useless.<\/span><\/span><\/p><p><span style=\"font-weight: 400; color: #000000;\">This loop happens when the substitution doesn&#8217;t simplify the fraction but instead recreates the same structure. Example:<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-651590b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"651590b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-02f9911\" data-id=\"02f9911\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e653f61 elementor-widget elementor-widget-text-editor\" data-id=\"e653f61\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h4 style=\"text-align: center;\"><span style=\"color: #800080;\"><b>\u222b[0 to \u03c0]\u00a0 x \u00b7 sin(x) \/ (1 + cos\u00b2x)\u00a0 dx<\/b><\/span><\/h4>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-36b0f6d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"36b0f6d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-147c0d6\" data-id=\"147c0d6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4fc63f3 elementor-widget elementor-widget-text-editor\" data-id=\"4fc63f3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400; color: #000000;\">Applying King&#8217;s rule integration here gives you a new expression that, when added to the original, doesn&#8217;t cancel cleanly. You go in circles. This is the exact moment the Queen&#8217;s rule of integration becomes essential.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-d7156bb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d7156bb\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-56d4634\" data-id=\"56d4634\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b8f4bee elementor-widget elementor-widget-text-editor\" data-id=\"b8f4bee\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #000000;\"><b>What Is Queen&#8217;s Rule in Integration?<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-ffbf130 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ffbf130\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cf22832\" data-id=\"cf22832\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1477715 elementor-widget elementor-widget-text-editor\" data-id=\"1477715\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Queen&#8217;s property of integration (also known as the Queen&#8217;s rule definite integration property) is a specialized tool designed for integrals whose upper limit can be written as <\/span><i><span style=\"font-weight: 400;\">2a<\/span><\/i><span style=\"font-weight: 400;\">. It works by checking the symmetry of the integrand around the midpoint <\/span><i><span style=\"font-weight: 400;\">a<\/span><\/i><span style=\"font-weight: 400;\">.<\/span><\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-08bc0c7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"08bc0c7\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-54c8a82\" data-id=\"54c8a82\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0349ab0 elementor-widget elementor-widget-text-editor\" data-id=\"0349ab0\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h4 style=\"text-align: center;\"><span style=\"color: #800080;\"><b>Queen&#8217;s Rule Formula: <\/b><span style=\"font-weight: 400;\">\u222b[0 to 2a] f(x) dx\u00a0 =\u00a0 \u222b[0 to a] [f(x) + f(2a \u2212 x)] dx<\/span><\/span><\/h4>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-5efa120 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5efa120\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4a16508\" data-id=\"4a16508\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c38a16b elementor-widget elementor-widget-text-editor\" data-id=\"c38a16b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #000000;\"><b>The Two Cases of Queen&#8217;s Property<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-c0c2e79 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c0c2e79\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ed82995\" data-id=\"ed82995\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-59f575b elementor-widget elementor-widget-text-editor\" data-id=\"59f575b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">The Queen&#8217;s property integration formula splits into two powerful outcomes based on the symmetry of <\/span><i><span style=\"font-weight: 400;\">f(2a \u2212 x)<\/span><\/i><span style=\"font-weight: 400;\">:<\/span><\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-cb86f43 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cb86f43\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9e8dc2d\" data-id=\"9e8dc2d\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-eef9927 elementor-widget elementor-widget-html\" data-id=\"eef9927\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<table style=\"border-collapse: collapse; width: 100%; font-family: Arial, sans-serif;\">\r\n  <thead>\r\n    <tr style=\"background-color:#4b5f7a; color:#ffffff;\">\r\n      <th style=\"padding:10px; border:1px solid #cfcfcf; text-align:left;\">Condition<\/th>\r\n      <th style=\"padding:10px; border:1px solid #cfcfcf; text-align:left;\">Result<\/th>\r\n    <\/tr>\r\n  <\/thead>\r\n  <tbody>\r\n    <tr style=\"background:#f5f5f5;\">\r\n      <td style=\"padding:10px; border:1px solid #cfcfcf;\">\r\n        f(2a \u2212 x) = f(x) &nbsp;&nbsp; [Symmetric]\r\n      <\/td>\r\n      <td style=\"padding:10px; border:1px solid #cfcfcf;\">\r\n        2 \u00d7 \u222b<sub>0<\/sub><sup>a<\/sup> f(x) dx\r\n      <\/td>\r\n    <\/tr>\r\n    <tr>\r\n      <td style=\"padding:10px; border:1px solid #cfcfcf;\">\r\n        f(2a \u2212 x) = \u2212f(x) <br \/>[Anti-symmetric]\r\n      <\/td>\r\n      <td style=\"padding:10px; border:1px solid #cfcfcf;\">\r\n        0\r\n      <\/td>\r\n    <\/tr>\r\n  <\/tbody>\r\n<\/table>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-5e54a28 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5e54a28\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c59dd6f\" data-id=\"c59dd6f\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-894c7c1 elementor-widget elementor-widget-text-editor\" data-id=\"894c7c1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">The key difference: while King&#8217;s rule of integration <\/span><i><span style=\"font-weight: 400;\">changes the integrand<\/span><\/i><span style=\"font-weight: 400;\">, the Queen&#8217;s rule of integration <\/span><i><span style=\"font-weight: 400;\">changes the limits<\/span><\/i><span style=\"font-weight: 400;\">, cutting the interval from [0, 2a] down to [0, a].<\/span><\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-51151bd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"51151bd\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-508b7d6\" data-id=\"508b7d6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dc93960 elementor-widget elementor-widget-text-editor\" data-id=\"dc93960\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #000000;\"><b>Solved Example: Using Queen&#8217;s Rule to Break the King&#8217;s Loop<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-e589caa elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e589caa\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-31744f9\" data-id=\"31744f9\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fad0403 elementor-widget elementor-widget-text-editor\" data-id=\"fad0403\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p style=\"text-align: center;\"><span style=\"font-weight: 400; color: #000000;\">Let&#8217;s solve the exact integral where King&#8217;s rule integration fails, and Queen&#8217;s rule integration saves the day.<\/span><\/p><table class=\" aligncenter\"><tbody><tr><td><p><span style=\"color: #000000;\"><b>Evaluate:\u00a0 I = \u222b[0 to \u03c0]\u00a0 x \/ (1 + sin x)\u00a0 dx<\/b><\/span><\/p><\/td><\/tr><\/tbody><\/table><h5 style=\"text-align: center;\"><span style=\"color: #008080;\"><b>Step 1 \u2014 Attempt King&#8217;s Rule (and See Why It Loops)<\/b><\/span><\/h5><p style=\"text-align: center;\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Applying <\/span>King&#8217;s rule integration<span style=\"font-weight: 400;\"> with <\/span><i><span style=\"font-weight: 400;\">a = 0, b = \u03c0<\/span><\/i><span style=\"font-weight: 400;\">, substitute <\/span><i><span style=\"font-weight: 400;\">x \u2192 (\u03c0 \u2212 x)<\/span><\/i><span style=\"font-weight: 400;\">:<\/span><\/span><\/p><table class=\" aligncenter\"><tbody><tr><td><p><span style=\"color: #000000;\"><b>I = \u222b[0 to \u03c0]\u00a0 (\u03c0 \u2212 x) \/ (1 + sin(\u03c0 \u2212 x))\u00a0 dx\u00a0 =\u00a0 \u222b[0 to \u03c0]\u00a0 (\u03c0 \u2212 x) \/ (1 + sin x)\u00a0 dx<\/b><\/span><\/p><\/td><\/tr><\/tbody><\/table><p style=\"text-align: center;\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Adding <\/span><i><span style=\"font-weight: 400;\">I + I<\/span><\/i><span style=\"font-weight: 400;\"> gives <\/span><i><span style=\"font-weight: 400;\">2I = \u03c0 \u00d7 \u222b[0 to \u03c0] 1\/(1 + sin x) dx<\/span><\/i><span style=\"font-weight: 400;\">. This <\/span><i><span style=\"font-weight: 400;\">does<\/span><\/i><span style=\"font-weight: 400;\"> work in this case, but for harder variants (like <\/span><i><span style=\"font-weight: 400;\">x\u00b7sin(x)\/(1 + cos\u00b2x)<\/span><\/i><span style=\"font-weight: 400;\">), the King loops. This is where the Queen&#8217;s property of integration takes over.<\/span><\/span><\/p><h5 style=\"text-align: center;\"><span style=\"color: #008080;\"><b>Classic Queen&#8217;s Rule Example: \u222b[0 to \u03c0] x\u00b7sin(x)\/(1 + cos\u00b2x) dx<\/b><\/span><\/h5><table class=\" aligncenter\"><tbody><tr><td><p><span style=\"color: #000000;\"><b>I = \u222b[0 to \u03c0]\u00a0 x \u00b7 sin(x) \/ (1 + cos\u00b2x)\u00a0 dx<\/b><\/span><\/p><\/td><\/tr><\/tbody><\/table><h5 style=\"text-align: center;\"><span style=\"color: #993366;\"><b>Step 1: Identify the Structure<\/b><\/span><\/h5><p style=\"text-align: center;\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Upper limit = \u03c0, so <\/span><b>2a = \u03c0 \u2192 a = \u03c0\/2<\/b><span style=\"font-weight: 400;\">. The integrand is <\/span><i><span style=\"font-weight: 400;\">f(x) = x \u00b7 sin(x) \/ (1 + cos\u00b2x)<\/span><\/i><span style=\"font-weight: 400;\">.<\/span><\/span><\/p><h5 style=\"text-align: center;\"><span style=\"color: #993366;\"><b>Step 2: Apply King&#8217;s Rule First<\/b><\/span><\/h5><p style=\"text-align: center;\"><span style=\"font-weight: 400; color: #000000;\">Substitute x \u2192 (\u03c0 \u2212 x) using King&#8217;s rule integration:<\/span><\/p><table class=\" aligncenter\"><tbody><tr><td><p><span style=\"color: #000000;\"><b>I = \u222b[0 to \u03c0]\u00a0 (\u03c0 \u2212 x) \u00b7 sin(\u03c0 \u2212 x) \/ (1 + cos\u00b2(\u03c0 \u2212 x))\u00a0 dx<\/b><\/span><\/p><\/td><\/tr><\/tbody><\/table><p style=\"text-align: center;\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Since <\/span><i><span style=\"font-weight: 400;\">sin(\u03c0 \u2212 x) = sin x<\/span><\/i><span style=\"font-weight: 400;\"> and <\/span><i><span style=\"font-weight: 400;\">cos(\u03c0 \u2212 x) = \u2212cos x \u2192 cos\u00b2(\u03c0 \u2212 x) = cos\u00b2x<\/span><\/i><span style=\"font-weight: 400;\">:<\/span><\/span><\/p><table class=\" aligncenter\"><tbody><tr><td><p><span style=\"color: #000000;\"><b>I = \u222b[0 to \u03c0]\u00a0 (\u03c0 \u2212 x) \u00b7 sin(x) \/ (1 + cos\u00b2x)\u00a0 dx<\/b><\/span><\/p><\/td><\/tr><\/tbody><\/table><h5 style=\"text-align: center;\"><span style=\"color: #993366;\"><b>Step 3: Add Both Forms of I<\/b><\/span><\/h5><table class=\" aligncenter\"><tbody><tr><td><p><span style=\"color: #000000;\"><b>2I = \u03c0 \u00b7 \u222b[0 to \u03c0]\u00a0 sin(x) \/ (1 + cos\u00b2x)\u00a0 dx<\/b><\/span><\/p><\/td><\/tr><\/tbody><\/table><h5 style=\"text-align: center;\"><span style=\"color: #993366;\"><b>Step 4: Now Apply Queen&#8217;s Property (Halve the Limit)<\/b><\/span><\/h5><p style=\"text-align: center;\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Since <\/span><i><span style=\"font-weight: 400;\">sin(\u03c0 \u2212 x)\/1+cos\u00b2(\u03c0 \u2212 x) = sin(x)\/1+cos\u00b2(x)<\/span><\/i><span style=\"font-weight: 400;\"> (i.e., <\/span><i><span style=\"font-weight: 400;\">f(2a \u2212 x) = f(x)<\/span><\/i><span style=\"font-weight: 400;\">), the <\/span><b>Queen&#8217;s rule definite integration<\/b><span style=\"font-weight: 400;\"> doubles it:<\/span><\/span><\/p><table class=\" aligncenter\"><tbody><tr><td><p><span style=\"color: #000000;\"><b>2I = 2\u03c0 \u00b7 \u222b[0 to \u03c0\/2]\u00a0 sin(x) \/ (1 + cos\u00b2x)\u00a0 dx<\/b><\/span><\/p><\/td><\/tr><\/tbody><\/table><h5 style=\"text-align: center;\"><span style=\"color: #993366;\"><b>Step 5: Evaluate with Substitution<\/b><\/span><\/h5><p style=\"text-align: center;\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Let <\/span><i><span style=\"font-weight: 400;\">u = cos x \u2192 du = \u2212sin x dx<\/span><\/i><span style=\"font-weight: 400;\">. When <\/span><i><span style=\"font-weight: 400;\">x = 0, u = 1<\/span><\/i><span style=\"font-weight: 400;\">; when <\/span><i><span style=\"font-weight: 400;\">x = \u03c0\/2, u = 0<\/span><\/i><span style=\"font-weight: 400;\">:<\/span><\/span><\/p><table class=\" aligncenter\"><tbody><tr><td><p><span style=\"color: #000000;\"><b>I = \u03c0 \u00b7 \u222b[0 to 1]\u00a0 du \/ (1 + u\u00b2)\u00a0 =\u00a0 \u03c0 \u00b7 [arctan u] from 0 to 1\u00a0 =\u00a0 \u03c0 \u00b7 \u03c0\/4\u00a0 =\u00a0 \u03c0\u00b2\/4<\/b><\/span><\/p><\/td><\/tr><\/tbody><\/table><p style=\"text-align: center;\">\u00a0<\/p><h5 style=\"text-align: center;\"><span style=\"color: #008000;\"><b>\u2713\u00a0 Answer: I = \u03c0\u00b2\/4<\/b><\/span><\/h5>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-e3f04b8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e3f04b8\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-361f14b\" data-id=\"361f14b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-831d57d elementor-widget elementor-widget-text-editor\" data-id=\"831d57d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #000000;\"><b>King&#8217;s Rule vs Queen&#8217;s Rule: Key Differences<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-82852d1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"82852d1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e8eb30b\" data-id=\"e8eb30b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-85a6c3f elementor-widget elementor-widget-html\" data-id=\"85a6c3f\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<table style=\"border-collapse: collapse; width: 100%; font-family: Arial, sans-serif;\">\r\n  <thead>\r\n    <tr style=\"background-color:#2f3142; color:#ffffff;\">\r\n      <th style=\"padding:10px; border:1px solid #bfc3c9; text-align:left;\">Feature<\/th>\r\n      <th style=\"padding:10px; border:1px solid #bfc3c9; text-align:left;\">King's Rule Integration<\/th>\r\n      <th style=\"padding:10px; border:1px solid #bfc3c9; text-align:left;\">Queen's Rule Integration<\/th>\r\n    <\/tr>\r\n  <\/thead>\r\n  <tbody>\r\n    <tr style=\"background:#d9d9d9;\">\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\"><strong>What it changes<\/strong><\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">The integrand (substitutes x)<\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">The limits (halves the interval)<\/td>\r\n    <\/tr>\r\n    <tr style=\"background:#efefef;\">\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\"><strong>Key substitution<\/strong><\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">x \u2192 a + b \u2212 x<\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">Checks f(2a \u2212 x) symmetry<\/td>\r\n    <\/tr>\r\n    <tr style=\"background:#d9d9d9;\">\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\"><strong>Best used when<\/strong><\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">Integrand simplifies with substitution<\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">King's rule loops; limit is 2a<\/td>\r\n    <\/tr>\r\n    <tr style=\"background:#efefef;\">\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\"><strong>Outcome<\/strong><\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">Adds two forms of I<\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">Doubles or zeroes the integral<\/td>\r\n    <\/tr>\r\n    <tr style=\"background:#d9d9d9;\">\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\"><strong>Interval<\/strong><\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">Any [a, b]<\/td>\r\n      <td style=\"padding:10px; border:1px solid #bfc3c9;\">Must be [0, 2a]<\/td>\r\n    <\/tr>\r\n  <\/tbody>\r\n<\/table>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-debf379 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"debf379\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3a67d64\" data-id=\"3a67d64\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6a87820 elementor-widget elementor-widget-text-editor\" data-id=\"6a87820\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3 style=\"text-align: center;\"><span style=\"color: #000000;\"><b>Step-by-Step Decision Framework<\/b><\/span><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-c14f0a7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c14f0a7\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ae7473a\" data-id=\"ae7473a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-72ff80d elementor-widget elementor-widget-text-editor\" data-id=\"72ff80d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400; color: #000000;\">When you face a complex definite integral, use this mental checklist:<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Identify the limits \u2014 is the upper limit of the form 2a (like \u03c0, 2, 4)?<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Try King&#8217;s rule integration first \u2014 substitute x \u2192 (a + b \u2212 x) and see if the sum simplifies.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Check if it loops \u2014 if adding the two forms gives I = I or doesn&#8217;t resolve, stop.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Switch to Queen&#8217;s rule of integration \u2014 test f(2a \u2212 x) for symmetry.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">If f(2a \u2212 x) = f(x): double and halve the limit. If f(2a \u2212 x) = \u2212f(x): the answer is 0.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Apply King&#8217;s rule again on the reduced integral if needed \u2014 now it often works perfectly.<\/span><\/li><\/ol><h4><span style=\"color: #000000;\"><b>Common Mistakes to Avoid<\/b><\/span><\/h4><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Skipping the symmetry test: Always verify whether f(2a \u2212 x) equals f(x) or \u2212f(x) before applying Queen&#8217;s rule for definite integration.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Forgetting to halve the limit: Queen&#8217;s property integration doubles the integral AND halves the upper limit. Both changes happen together.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Applying Queen&#8217;s rule when limits aren&#8217;t [0, 2a]: This property only works for integrals starting at 0.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Using King&#8217;s rule when the substitution creates an unsolvable loop: Recognize the loop early and switch strategies.<\/span><\/li><\/ul>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-000fa3b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"000fa3b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1901152\" data-id=\"1901152\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-efb1fbf elementor-widget elementor-widget-text-editor\" data-id=\"efb1fbf\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><span style=\"color: #008000;\"><b>A Tip for SAT prep<\/b><\/span><\/h3><p><span style=\"font-weight: 400;\">For those of you currently grinding through SAT prep, remember: math isn&#8217;t just about memorizing one way to do things. It\u2019s about building a &#8220;toolbox.&#8221; The King\u2019s Rule is great, but the Queen\u2019s Rule is the specialized tool you pull out when things get really interesting.<\/span><\/p><p><b>Recommended Reading:<\/b><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/exponential-equations-using-recursion-and-algebraic\/\"><span style=\"font-weight: 400;\">Solving Exponential Equations Using Recursion: A Step-by-Step Guide<\/span><\/a><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/linear-equations-different-solutions\/\"><span style=\"font-weight: 400;\">Linear Equation &#8211; One Solution, No Solution and Many Solutions<\/span><\/a><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/geometry-problem\/\"><span style=\"font-weight: 400;\">Interesting Geometry Problem to Solve For Kids<\/span><\/a><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/sat-quadratics-tricks\/\"><span style=\"font-weight: 400;\">The Ultimate Guide to Solving SAT Quadratics in Seconds<\/span><\/a><\/p><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/derive-quadratic-formula\/\"><span style=\"font-weight: 400;\">How to Derive and Use the Quadratic Formula (With Examples)<\/span><\/a><\/p><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/sherman-morrison-woodbury-identity\/\"><span style=\"font-weight: 400;\">Application &amp; Proof\u00a0 of the Sherman-Morrison-Woodbury Identity<\/span><\/a><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/geometric-problem-unsolved-by-ai\/\"><span style=\"font-weight: 400;\">The Geometry Problem That Still Defeats ChatGPT, Gemini, and Grok<\/span><\/a><\/li><\/ol><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/wp-content\/uploads\/2026\/02\/queens_property.jpeg\"><img decoding=\"async\" loading=\"lazy\" class=\"alignnone size-full wp-image-37963\" src=\"https:\/\/mp.moonpreneur.com\/math-corner\/wp-content\/uploads\/2026\/02\/queens_property.jpeg\" alt=\"\" width=\"1935\" height=\"1080\" \/><\/a><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-4dd7e1e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4dd7e1e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-70c82a2\" data-id=\"70c82a2\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-eb6ea7a elementor-widget elementor-widget-text-editor\" data-id=\"eb6ea7a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><strong>FAQs on\u00a0 Queen&#8217;s Property<\/strong><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-44c18e9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"44c18e9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-24f8067\" data-id=\"24f8067\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b4d1436 elementor-widget elementor-widget-elementskit-faq\" data-id=\"b4d1436\" data-element_type=\"widget\" data-widget_type=\"elementskit-faq.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div class=\"ekit-wid-con\">\n                <div class=\"elementskit-single-faq elementor-repeater-item-95228b5\">\n            <div class=\"elementskit-faq-header\">\n                <h2 class=\"elementskit-faq-title\">Q1. What is the main purpose of Queen&#039;s rule integration?<\/h2>\n            <\/div>\n            <div class=\"elementskit-faq-body\">\n                Ans. Queen's rule integration is a specialized tool that simplifies complex integrals by checking the symmetry of the integrand around the midpoint of the interval. It is specifically designed to break the loops that occur when King's rule integration leads you back to the starting point without solving the problem.\n\n            <\/div>\n        <\/div>\n                <div class=\"elementskit-single-faq elementor-repeater-item-62a8206\">\n            <div class=\"elementskit-faq-header\">\n                <h2 class=\"elementskit-faq-title\">Q2. When should I use Queen&#039;s rule instead of King&#039;s rule integration?<\/h2>\n            <\/div>\n            <div class=\"elementskit-faq-body\">\n                Ans. Switch to Queen's property of integration when applying King's rule integration results in I = I, meaning the integral equals itself, and no new information is gained. Also, use it whenever the upper limit is exactly 2a, and you suspect symmetry in the integrand.\n\n            <\/div>\n        <\/div>\n                <div class=\"elementskit-single-faq elementor-repeater-item-80e5d28\">\n            <div class=\"elementskit-faq-header\">\n                <h2 class=\"elementskit-faq-title\">Q3. How does Queen&#039;s rule change the limits of integration?<\/h2>\n            <\/div>\n            <div class=\"elementskit-faq-body\">\n                Ans. The Queen's rule definite integration property takes the original upper limit (2a) and cuts it exactly in half (to a). In exchange, the integrand becomes the sum of f(x) and f(2a \u2212 x). If these are equal, the result doubles; if they are negatives of each other, the result is zero.\n            <\/div>\n        <\/div>\n                <div class=\"elementskit-single-faq elementor-repeater-item-33ce851\">\n            <div class=\"elementskit-faq-header\">\n                <h2 class=\"elementskit-faq-title\">Q4. Can King&#039;s rule and Queen&#039;s rule be used together?<\/h2>\n            <\/div>\n            <div class=\"elementskit-faq-body\">\n                Ans. Yes, and that combination is often the most powerful approach. A common strategy is to apply King's rule integration first to simplify the numerator (e.g., converting x into \u03c0 \u2212 x), then use Queen's property integration to halve the resulting limits and evaluate the integral cleanly.\n            <\/div>\n        <\/div>\n                <div class=\"elementskit-single-faq elementor-repeater-item-e0524b5\">\n            <div class=\"elementskit-faq-header\">\n                <h2 class=\"elementskit-faq-title\">Q5. Is Queen&#039;s rule the same as the property for even\/odd functions?<\/h2>\n            <\/div>\n            <div class=\"elementskit-faq-body\">\n                Ans. They are related but not identical. The even\/odd property applies to integrals over symmetric intervals like [\u2212a, a], while Queen's rule of integration specifically targets integrals over [0, 2a]. Both exploit function symmetry, but the formulas and conditions differ.\n            <\/div>\n        <\/div>\n        \n    <\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-f492ea9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f492ea9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-af23f37\" data-id=\"af23f37\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d14873f elementor-widget elementor-widget-text-editor\" data-id=\"d14873f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h5><span style=\"color: #800000;\"><b>Pro Tip for Exam Day<\/b><\/span><\/h5><p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Think of King&#8217;s rule integration and Queen&#8217;s rule integration as a team. The King transforms the integrand; the Queen transforms the limits. When one hits a wall, the other opens a door. Mastering both, and knowing <\/span><i><span style=\"font-weight: 400;\">when<\/span><\/i><span style=\"font-weight: 400;\"> to switch, is what separates students who struggle from students who solve any integral with confidence.<\/span><\/span><\/p><p><span style=\"font-weight: 400; color: #000000;\">Want to excite your child about math and sharpen their math skills? Moonpreneur&#8217;s online math curriculum is unique as it helps children understand math skills through hands-on lessons, assists them in building real-life applications, and excites them to learn math.\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\"><span style=\"color: #000000;\">You can opt for our<\/span> <\/span><a href=\"https:\/\/moonpreneur.com\/innovator-program\/advanced-math\/\"><span style=\"font-weight: 400;\">Advanced Math<\/span><\/a><span style=\"font-weight: 400;\"><span style=\"color: #000000;\"> or Vedic Math+Mental Math courses. Our<\/span> <\/span><a href=\"https:\/\/mp.moonpreneur.com\/math-quiz-for-kids\/\"><span style=\"font-weight: 400;\">Math Quiz<\/span><\/a><span style=\"font-weight: 400; color: #000000;\"> for grades 3rd, 4th, 5th, and 6th helps in further exciting and engaging in mathematics with hands-on lessons.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Update: This article was last updated on 12th March 2026 to reflect the accuracy and up-to-date information on the page. You&#8217;re staring at a definite integral packed with sines, cosines, and fractions. It looks impossible. Your first instinct? Reach for a calculator. But before you do, what if there was a two-step royal shortcut that [&hellip;]<\/p>\n","protected":false},"author":116,"featured_media":38116,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[979,1034,986],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/37955"}],"collection":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/users\/116"}],"replies":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/comments?post=37955"}],"version-history":[{"count":14,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/37955\/revisions"}],"predecessor-version":[{"id":38154,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/37955\/revisions\/38154"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media\/38116"}],"wp:attachment":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media?parent=37955"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/categories?post=37955"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/tags?post=37955"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}