{"id":38566,"date":"2026-05-27T13:29:29","date_gmt":"2026-05-27T13:29:29","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=38566"},"modified":"2026-05-28T14:03:06","modified_gmt":"2026-05-28T14:03:06","slug":"integration-of-cosec-x-formula","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/integration-of-cosec-x-formula\/","title":{"rendered":"What is the Integration of Cosec X?"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"38566\" class=\"elementor elementor-38566\" 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-5183dc2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5183dc2\" 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-d56c2b9\" data-id=\"d56c2b9\" 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-330bc4d elementor-widget elementor-widget-html\" data-id=\"330bc4d\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div class=\"math-card\">\r\n\r\n  <h2>Integration of Cosec x<\/h2>\r\n\r\n  <input type=\"range\" min=\"1\" max=\"10\" value=\"2\" class=\"slider\" id=\"angleSlider\"\/>\r\n\r\n  <div class=\"circle-box\">\r\n    <div class=\"circle\">\r\n\r\n      <div class=\"radius-line\"><\/div>\r\n      <div class=\"center-dot\"><\/div>\r\n\r\n      <div class=\"radius-text\">\r\n        x = <span id=\"xVal\">2<\/span>\r\n      <\/div>\r\n\r\n      <div class=\"area-text\">\r\n        \u222bcscx dx\r\n      <\/div>\r\n\r\n    <\/div>\r\n  <\/div>\r\n\r\n  <table>\r\n    <tr>\r\n      <th>Formula<\/th>\r\n      <td>\u222b cscx dx = ln|cscx \u2212 cotx| + C<\/td>\r\n    <\/tr>\r\n\r\n    <tr>\r\n      <th>Angle (x)<\/th>\r\n      <td><span id=\"tableX\">2<\/span> rad<\/td>\r\n    <\/tr>\r\n\r\n    <tr>\r\n      <th>Result Form<\/th>\r\n      <td>ln|csc(<span id=\"tableX2\">2<\/span>) \u2212 cot(<span id=\"tableX3\">2<\/span>)| + C<\/td>\r\n    <\/tr>\r\n  <\/table>\r\n\r\n<\/div>\r\n\r\n<style>\r\n.math-card{\r\n  max-width:500px;\r\n  margin:auto;\r\n  background:#fff;\r\n  padding:20px;\r\n  border-radius:18px;\r\n  box-shadow:0 10px 25px rgba(0,0,0,0.08);\r\n  font-family:Arial,sans-serif;\r\n}\r\n\r\n.slider{\r\n  width:100%;\r\n  margin-bottom:20px;\r\n}\r\n\r\n.circle-box{\r\n  display:flex;\r\n  justify-content:center;\r\n  margin-bottom:20px;\r\n}\r\n\r\n.circle{\r\n  width:180px;\r\n  height:180px;\r\n  border:5px solid #ef4444;\r\n  border-radius:50%;\r\n  position:relative;\r\n  background:rgba(239,68,68,0.08);\r\n}\r\n\r\n.radius-line{\r\n  width:70px;\r\n  height:4px;\r\n  background:#22c55e;\r\n  position:absolute;\r\n  top:50%;\r\n  left:50%;\r\n  transform-origin:left center;\r\n  animation:spin 5s linear infinite;\r\n}\r\n\r\n.center-dot{\r\n  width:14px;\r\n  height:14px;\r\n  background:#2563eb;\r\n  border-radius:50%;\r\n  position:absolute;\r\n  top:50%;\r\n  left:50%;\r\n  transform:translate(-50%,-50%);\r\n}\r\n\r\n.radius-text{\r\n  position:absolute;\r\n  top:40%;\r\n  left:58%;\r\n  color:#22c55e;\r\n  font-weight:bold;\r\n}\r\n\r\n.area-text{\r\n  position:absolute;\r\n  bottom:-28px;\r\n  left:35px;\r\n  color:#7c3aed;\r\n  font-weight:bold;\r\n}\r\n\r\ntable{\r\n  width:100%;\r\n  border-collapse:collapse;\r\n}\r\n\r\ntable th,\r\ntable td{\r\n  border:1px solid #ddd;\r\n  padding:12px;\r\n}\r\n\r\ntable th{\r\n  background:#f3f4f6;\r\n}\r\n\r\n@keyframes spin{\r\n  from{transform:rotate(0deg);}\r\n  to{transform:rotate(360deg);}\r\n}\r\n<\/style>\r\n\r\n<script>\r\nconst slider = document.getElementById(\"angleSlider\");\r\n\r\nslider.oninput = function(){\r\n\r\n  let x = this.value;\r\n\r\n  document.getElementById(\"xVal\").innerHTML = x;\r\n  document.getElementById(\"tableX\").innerHTML = x;\r\n  document.getElementById(\"tableX2\").innerHTML = x;\r\n  document.getElementById(\"tableX3\").innerHTML = x;\r\n};\r\n<\/script>\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-2c3b27e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2c3b27e\" 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-39930b5\" data-id=\"39930b5\" 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-55e334a elementor-widget elementor-widget-text-editor\" data-id=\"55e334a\" 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;\">Imagine you&#8217;re at an amusement park, riding a roller coaster. As you go up and down, your height keeps changing smoothly\u2014 just like a sine wave. Now flip that idea upside down&#8230; and suddenly, you&#8217;re dealing with cosec x!<\/span><\/p><p><span style=\"font-weight: 400;\">That&#8217;s where things get exciting. When students first encounter the cosec x function, it can feel tricky\u2014almost like a loop-the-loop on that roller coaster. But once you understand the pattern, it becomes surprisingly satisfying.<\/span><\/p><p><b><i>Let&#8217;s understand the i<\/i><\/b><b>ntegration of cosec x <\/b><b><i>in depth.<\/i><\/b><\/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-d4b8b61 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d4b8b61\" 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-14d66bb\" data-id=\"14d66bb\" 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-38c18af elementor-widget elementor-widget-text-editor\" data-id=\"38c18af\" 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<h2><b>I<\/b><b>ntegration of cosec x<\/b><b> (Compl<\/b><b>ete Guide)<\/b><\/h2>\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-10aa652 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"10aa652\" 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-b97e71f\" data-id=\"b97e71f\" 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-146543a elementor-widget elementor-widget-text-editor\" data-id=\"146543a\" 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=\"font-weight: 400;\"><b>What is Cosec X?<\/b><\/span><\/h3><p><span style=\"font-weight: 400;\">Before diving into integration, let&#8217;s recall:<\/span><\/p><p><span style=\"font-weight: 400;\">cosec x=1\/sin x\u00a0\u00a0\u00a0<\/span><\/p><p><span style=\"font-weight: 400;\">It&#8217;s simply the reciprocal of sine, and plays a major role in trigonometric calculus.<\/span><\/p><h3><b>Primary Concepts: integration of cosec x <\/b><\/h3><p><span style=\"font-weight: 400;\">Now, here comes the star of the show: \u222bcsc x dx = ln|csc x\u2212cot x |+C<\/span><\/p><p><span style=\"font-weight: 400;\">This is the standard formula for the <\/span><b>integration of cosec x.<\/b><\/p><p><span style=\"font-weight: 400;\">Where:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">In = Natural logarithm<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">C = Constant of integration<\/span><\/li><\/ul><h3><b>Equivalent Forms of the Formula<\/b><\/h3><p><span style=\"font-weight: 400;\">In addition, the integration of cosec x can also be written in different but equivalent forms:<\/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-5935292 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5935292\" 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-7b47d1b\" data-id=\"7b47d1b\" 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-b152bf9 elementor-widget elementor-widget-html\" data-id=\"b152bf9\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div style=\"max-width:650px;margin:20px auto;font-family:Arial;border-radius:10px;overflow:hidden;box-shadow:0 4px 10px rgba(0,0,0,0.1);\">\r\n\r\n  <h3 style=\"background:#6c63ff;color:#fff;text-align:center;padding:10px;margin:0;\">\r\n    Equivalent Forms of \u222bcscx dx\r\n  <\/h3>\r\n\r\n  <table style=\"width:100%;border-collapse:collapse;background:#f9f9ff;font-size:15px;\">\r\n    \r\n    <tr style=\"background:#e6e6ff;\">\r\n      <th style=\"padding:10px;border:1px solid #ddd;\">Form<\/th>\r\n      <th style=\"padding:10px;border:1px solid #ddd;\">Formula<\/th>\r\n    <\/tr>\r\n\r\n    <tr>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">Form 1<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\r\n        \u222bcscxdx = \u222bcscxdx = \u2212ln\u2223cscx+cotx\u2223 + C\r\n      <\/td>\r\n    <\/tr>\r\n\r\n    <tr style=\"background:#f1f1ff;\">\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">Form 2<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\r\n        \u222bcscxdx = \u222bcscxdx = ln\u2223tan(x\/2)\u2223 + C\r\n      <\/td>\r\n    <\/tr>\r\n\r\n    <tr>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">Form 3<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\r\n        \u222bcscxdx = 21\u200bln (1\u2212cosx \/ 1+cosx) + C\r\n      <\/td>\r\n    <\/tr>\r\n\r\n  <\/table>\r\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-da6b880 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"da6b880\" 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-c9bd08e\" data-id=\"c9bd08e\" 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-788cd81 elementor-widget elementor-widget-text-editor\" data-id=\"788cd81\" 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;\">These variations are useful depending on the type of problem you&#8217;re solving<\/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-44964ea elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"44964ea\" 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-d74ad00\" data-id=\"d74ad00\" 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-168d7f0 elementor-widget elementor-widget-text-editor\" data-id=\"168d7f0\" 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><b>Other Important Integration\u00a0 Form<\/b><\/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-9f304e5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9f304e5\" 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-07c3ff2\" data-id=\"07c3ff2\" 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-9d06e6e elementor-widget elementor-widget-text-editor\" data-id=\"9d06e6e\" 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>Now, let&#8217;s look at other important forms.<\/p><p><b>Integration of cosec ^2x\/integration of cosec sqaure x<\/b><\/p><p><span style=\"font-weight: 400;\">\u222bcsc2xdx=\u2212cotx+C<\/span><\/p><p><b>Integration of cosec x cot x<\/b><\/p><p><span style=\"font-weight: 400;\">\u222bcscxcotxdx=\u2212cscx+C<\/span><\/p><p><span style=\"font-weight: 400;\">This is a direct derivative-based result and a very important exam.<\/span><\/p><p><b>Integration of cosec inverse <\/b><span style=\"font-weight: 400;\">x<\/span><\/p><p>\u222bcsc\u22121xdx=xcsc\u22121x+ln\u200bx+x2\u22121\u200b\u200b+C<\/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-904846d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"904846d\" 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-b6312a5\" data-id=\"b6312a5\" 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-8296a6c elementor-widget elementor-widget-text-editor\" data-id=\"8296a6c\" 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 data-section-id=\"1atau6u\" data-start=\"1047\" data-end=\"1086\"><b>How is the Integration of cosec x derived?<\/b><\/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-cf94d30 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cf94d30\" 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-68bb32b\" data-id=\"68bb32b\" 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-08cb8c7 elementor-widget elementor-widget-text-editor\" data-id=\"08cb8c7\" 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;\">Now Here&#8217;s the trick(the &#8220;magic step&#8221; students love):<\/span><\/p><p><span style=\"font-weight: 400;\">We multiply and divide by:<\/span><\/p><p><span style=\"font-weight: 400;\">(cscx\u2212cotx)<\/span><\/p><p><span style=\"font-weight: 400;\">This transforms the integral into a logarithmic derivative form, making it easy to solve. The idea is to force the numerator to look like a derivative of the denominator<\/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-37875b1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"37875b1\" 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-8cab757\" data-id=\"8cab757\" 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-267e908 elementor-widget elementor-widget-text-editor\" data-id=\"267e908\" 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><b>Methods to Solve<\/b><\/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-de3e8f9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"de3e8f9\" 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-ac5c8af\" data-id=\"ac5c8af\" 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-a584023 elementor-widget elementor-widget-text-editor\" data-id=\"a584023\" 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>There are mainly two methods to solve this:<\/p><ol><li><b> Substitution method<\/b><\/li><\/ol><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Let:u=cscx\u2212cotx<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Convert into: \u222bdu\/u<\/span><\/li><\/ul><ol start=\"2\"><li><b> Using Identities<\/b><\/li><\/ol><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Convert into sine\/cosine form<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Use identities like: <\/span>1\u2212cos2x=sin2x<\/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-d9cd243 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d9cd243\" 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-86a95cd\" data-id=\"86a95cd\" 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-d093163 elementor-widget elementor-widget-text-editor\" data-id=\"d093163\" 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><b>Applications of <\/b><b>the integration of cosec x<\/b><b>\u00a0<\/b><\/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-81974e2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"81974e2\" 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-c5a652c\" data-id=\"c5a652c\" 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-ecae0f7 elementor-widget elementor-widget-text-editor\" data-id=\"ecae0f7\" 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;\">This concept is just a theory\u2014it&#8217;s used in:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Physics: Wave motion, oscillations<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Electrical Engineering: AC Circuits, phase angles<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">mathematics: Solving differential equations<\/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-4ec8a6e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4ec8a6e\" 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-5fa7ae8\" data-id=\"5fa7ae8\" 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-86cd43e elementor-widget elementor-widget-text-editor\" data-id=\"86cd43e\" 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><b>Definite Integral of Cosec x<\/b><\/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-9324540 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9324540\" 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-8948ad8\" data-id=\"8948ad8\" 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-ba98090 elementor-widget elementor-widget-text-editor\" data-id=\"ba98090\" 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>Now, let&#8217;s look at this:<\/p><p><span style=\"font-weight: 400;\">\u222bab\u200bcscxdx=[ln\u2223cscx\u2212cotx\u2223]a\u00a0<\/span><\/p><p><b>Apply upper and lower limits:<\/b><\/p><p><span style=\"font-weight: 400;\">=ln|csc(b)\u2212cot(b)|\u2212ln\u2223csc(a)\u2212cot(a)|<\/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-8f7fc22 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8f7fc22\" 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-a485518\" data-id=\"a485518\" 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-ccc7700 elementor-widget elementor-widget-text-editor\" data-id=\"ccc7700\" 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>Examples:<\/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-094b318 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"094b318\" 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-1ff1953\" data-id=\"1ff1953\" 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-c1ca8d7 elementor-widget elementor-widget-html\" data-id=\"c1ca8d7\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div style=\"max-width:650px;margin:20px auto;font-family:Arial;border-radius:10px;overflow:hidden;border:1px solid #ddd;\">\r\n  \r\n  <h3 style=\"text-align:center;background:#6c63ff;color:#fff;padding:10px;margin:0;\">\r\n    Integration of Cosec x \u2013 Examples\r\n  <\/h3>\r\n\r\n  <table style=\"width:100%;border-collapse:collapse;font-size:15px;\">\r\n    \r\n    <tr style=\"background:#f3f4ff;\">\r\n      <th style=\"padding:10px;border:1px solid #ddd;\">Example<\/th>\r\n      <th style=\"padding:10px;border:1px solid #ddd;\">Integral Formula<\/th>\r\n    <\/tr>\r\n\r\n    <tr style=\"background:#ffffff;\">\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">Example 1<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\r\n        \u222bcscx dx = ln|cscx \u2212 cotx| + C\r\n      <\/td>\r\n    <\/tr>\r\n\r\n    <tr style=\"background:#f9f9ff;\">\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">Example 2<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\r\n        \u222bcsc\u00b2x dx = \u2212cotx + C\r\n      <\/td>\r\n    <\/tr>\r\n\r\n    <tr style=\"background:#ffffff;\">\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">Example 3<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\r\n        \u222bcscx cotx dx = \u2212cscx + C\r\n      <\/td>\r\n    <\/tr>\r\n\r\n  <\/table>\r\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-0804a3d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0804a3d\" 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-bef5ac9\" data-id=\"bef5ac9\" 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-6bb4b55 elementor-widget elementor-widget-text-editor\" data-id=\"6bb4b55\" 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><b>Common Mistakes Students Make<\/b><\/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-db8aa9f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"db8aa9f\" 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-899a9dc\" data-id=\"899a9dc\" 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-ac82ded elementor-widget elementor-widget-text-editor\" data-id=\"ac82ded\" 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<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Learning the Times Tables without Understanding<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Students should learn and understand the repeating patterns and how multiple times can be seen in addition.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Not Practicing on a Daily Basis<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">In order to remember any new information, it is important to review what has been learned regularly.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Learning All the Times Tables Together<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">It is recommended to learn two or three at a time.<\/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-14ceaa0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"14ceaa0\" 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-6782616\" data-id=\"6782616\" 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-6daccc0 elementor-widget elementor-widget-text-editor\" data-id=\"6daccc0\" 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><b>Here are some of the Practice Questions<\/b><\/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-662eaf2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"662eaf2\" 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-cb52f3b\" data-id=\"cb52f3b\" 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-82d9ff6 elementor-widget elementor-widget-text-editor\" data-id=\"82d9ff6\" 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>Now, try these yourself:<\/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-255da27 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"255da27\" 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-cd2af5a\" data-id=\"cd2af5a\" 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-597f7b8 elementor-widget elementor-widget-html\" data-id=\"597f7b8\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<div style=\"max-width:700px;margin:20px auto;font-family:Arial;border:1px solid #ddd;border-radius:10px;overflow:hidden;\">\r\n  \r\n  <h3 style=\"text-align:center;background:#ff7a18;color:#fff;padding:12px;margin:0;\">\r\n    Practice Questions \u2013 Integration of Cosec x\r\n  <\/h3>\r\n\r\n  <table style=\"width:100%;border-collapse:collapse;font-size:15px;\">\r\n    \r\n    <tr style=\"background:#fff3e6;\">\r\n      <th style=\"padding:10px;border:1px solid #ddd;\">Question No.<\/th>\r\n      <th style=\"padding:10px;border:1px solid #ddd;\">Problem<\/th>\r\n    <\/tr>\r\n\r\n    <tr style=\"background:#ffffff;\">\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">1<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\u222b csc\u00b2x dx<\/td>\r\n    <\/tr>\r\n\r\n    <tr style=\"background:#fff9f2;\">\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">2<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\u222b cscx cotx dx<\/td>\r\n    <\/tr>\r\n\r\n    <tr style=\"background:#ffffff;\">\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">3<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\r\n        \u222b<sub>\u03c0\/4<\/sub><sup>\u03c0\/2<\/sup> cscx dx\r\n      <\/td>\r\n    <\/tr>\r\n\r\n    <tr style=\"background:#fff9f2;\">\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">4<\/td>\r\n      <td style=\"padding:10px;border:1px solid #ddd;\">\r\n        Derive the formula of integration of cosec x\r\n      <\/td>\r\n    <\/tr>\r\n\r\n  <\/table>\r\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-dcd6bb6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"dcd6bb6\" 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-b4f4652\" data-id=\"b4f4652\" 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-21748f7 elementor-widget elementor-widget-text-editor\" data-id=\"21748f7\" 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><strong><em>Read more related articles:<\/em><\/strong><\/p><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/how-to-find-the-area-of-a-circle\/\">\u2705How to Find the Area of a Circle?<\/a><\/p><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/how-to-find-the-area-of-a-circle\/\">\u2705<\/a><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/vertex-formula-explained\/\">What is the Vertex Formula?<\/a><\/p><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/horizontal-asymptote\/\">\u2705Horizontal Asymptote: Rules, Formula, and Easy Examples<\/a><\/p><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/how-to-find-the-radius-of-a-circle\/\">\u2705How to Find the Radius of a Circle: Easy Formulas and Examples<\/a><\/p><p><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/wallis-formula-integration-sine-cosine\/\">\u2705The Wallis Formula: Integrating Powers of Sine and Cosine Instantly<\/a><\/p><p class=\"elementor-post__title\"><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/kings-rule-in-definite-integrals\/\">\u2705How to Use King\u2019s Rule in Definite Integrals: Formulas &amp; Solved Examples<\/a><\/p><p class=\"elementor-post__title\"><a href=\"https:\/\/mp.moonpreneur.com\/math-corner\/use-of-cosine-formula\/\">\u2705How to Use the Cosine Formula to Find Missing Sides and Angles<\/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-88bba79 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"88bba79\" 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-3fb2615\" data-id=\"3fb2615\" 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-abd5118 elementor-widget elementor-widget-text-editor\" data-id=\"abd5118\" 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><b>Conclusion<\/b><b><br \/><\/b><\/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-e80ae28 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e80ae28\" 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-3d21413\" data-id=\"3d21413\" 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-4e8e04c elementor-widget elementor-widget-text-editor\" data-id=\"4e8e04c\" 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;\">The integration of cosec x equals ln|csc x \u2212 cot x| + C and is derived using a smart algebraic trick. It related forms\u2014integration of cosec ^ 2 x, integration of cosec^2 x, integration of cosec x cot x, and integration of cosec inverse x\u2014are essential tools in calculus problem-solving.<\/span><\/p><p><span style=\"font-weight: 400;\">You can opt for our <a href=\"https:\/\/moonpreneur.com\/innovator-program\/advanced-math\/\">Advanced Math<\/a> or Vedic Math+Mental Math courses. Our <a href=\"https:\/\/mp.moonpreneur.com\/math-quiz-for-kids\/\">Math Quiz<\/a> for grades 3rd, 4th, 5th, and 6th helps in further exciting and engaging in mathematics with hands-on lessons.<br \/><\/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-f6e441a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f6e441a\" 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-f63cc91\" data-id=\"f63cc91\" 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-a14c32f elementor-widget elementor-widget-text-editor\" data-id=\"a14c32f\" 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<\/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-930af6e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"930af6e\" 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-756d14f\" data-id=\"756d14f\" 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-466481a elementor-widget elementor-widget-toggle\" data-id=\"466481a\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-7381\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-7381\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">1. What is the integration of cosec x?<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-7381\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-7381\"><p>Ans. <span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-symbol large-op\">\u222b<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mop\">ln<\/span><span class=\"minner\"><span class=\"mopen delimcenter\">\u2223<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose delimcenter\">\u2223<\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span><\/span><\/p><p data-start=\"171\" data-end=\"202\">\ud83d\udc49 This can also be written as:<\/p><p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u222b<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\">\u2223<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>+<\/mo><mi>cot<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo fence=\"true\">\u2223<\/mo><\/mrow><mo>+<\/mo><mi>C<\/mi><annotation encoding=\"application\/x-tex\">\\int \\csc x \\, dx = -\\ln \\left| \\csc x + \\cot x \\right| + C<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-symbol large-op\">\u222b<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mop\">ln<\/span><span class=\"minner\"><span class=\"mopen delimcenter\">\u2223<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose delimcenter\">\u2223<\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span><\/span><\/p><hr data-start=\"271\" data-end=\"274\" \/><p data-section-id=\"1ewwnaf\" data-start=\"276\" data-end=\"297\"><strong>Final Answer<\/strong><\/p><p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><menclose notation=\"box\"><mstyle displaystyle=\"false\" scriptlevel=\"0\"><\/mstyle><mstyle displaystyle=\"false\" scriptlevel=\"0\"><\/mstyle><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow><mo>\u222b<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\">\u2223<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mi>cot<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo fence=\"true\">\u2223<\/mo><\/mrow><mo>+<\/mo><mi>C<\/mi><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\int \\csc x \\, dx = \\ln \\left| \\csc x &#8211; \\cot x \\right| + C}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"boxpad\"><span class=\"mop op-symbol large-op\">\u222b<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><span class=\"mop\">ln<\/span><span class=\"minner\"><span class=\"mopen delimcenter\">\u2223<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose delimcenter\">\u2223<\/span><\/span><span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">C<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-7382\" class=\"elementor-tab-title\" data-tab=\"2\" role=\"button\" aria-controls=\"elementor-tab-content-7382\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">2. How to integrate cosec theta?<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-7382\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"2\" role=\"region\" aria-labelledby=\"elementor-tab-title-7382\"><p>Ans.<\/p><p data-section-id=\"1hfd1n9\" data-start=\"101\" data-end=\"121\">Final Formula<\/p><p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u222b<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>\u03b8<\/mi><mo>=<\/mo><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\">\u2223<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo>\u2212<\/mo><mi>cot<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo fence=\"true\">\u2223<\/mo><\/mrow><mo>+<\/mo><mi>C<\/mi><annotation encoding=\"application\/x-tex\">\\int \\csc \\theta \\, d\\theta = \\ln \\left| \\csc \\theta &#8211; \\cot \\theta \\right| + C<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-symbol large-op\">\u222b<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mop\">ln<\/span><span class=\"minner\"><span class=\"mopen delimcenter\">\u2223<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose delimcenter\">\u2223<\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span><\/span><\/p><p>\ud83d\udc49 Equivalent form:<\/p><p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u222b<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>\u03b8<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>ln<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo fence=\"true\">\u2223<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo>+<\/mo><mi>cot<\/mi><mo>\u2061<\/mo><mi>\u03b8<\/mi><mo fence=\"true\">\u2223<\/mo><\/mrow><mo>+<\/mo><mi>C<\/mi><annotation encoding=\"application\/x-tex\">\\int \\csc \\theta \\, d\\theta = -\\ln \\left| \\csc \\theta + \\cot \\theta \\right| + C<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-symbol large-op\">\u222b<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mop\">ln<\/span><span class=\"minner\"><span class=\"mopen delimcenter\">\u2223<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mbin\">+<\/span><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">\u03b8<\/span><span class=\"mclose delimcenter\">\u2223<\/span><\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span><\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-7383\" class=\"elementor-tab-title\" data-tab=\"3\" role=\"button\" aria-controls=\"elementor-tab-content-7383\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">3. What is integration of cosec x * cot x?<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-7383\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"3\" role=\"region\" aria-labelledby=\"elementor-tab-title-7383\"><p>Ans.<\/p><p data-start=\"0\" data-end=\"77\">To integrate <strong data-start=\"13\" data-end=\"42\"><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>\u22c5<\/mo><mi>cot<\/mi><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\csc x \\cdot \\cot x<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u22c5<\/span><\/span><span class=\"base\"><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/strong>, use a direct derivative identity.<\/p><hr data-start=\"79\" data-end=\"82\" \/><p data-section-id=\"1trjcly\" data-start=\"84\" data-end=\"105\">Key Identity<\/p><p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mi>cot<\/mi><mo>\u2061<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{d}{dx}(\\csc x) = -\\csc x \\cot x<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\">d<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/span><\/p><hr data-start=\"151\" data-end=\"154\" \/><p data-section-id=\"12ejzft\" data-start=\"156\" data-end=\"174\">Integration<\/p><p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u222b<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mi>cot<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\int \\csc x \\cot x \\, dx = -\\csc x + C<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-symbol large-op\">\u222b<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span><\/span><\/p><hr data-start=\"222\" data-end=\"225\" \/><p data-section-id=\"1ewwnaf\" data-start=\"227\" data-end=\"248\">Final Answer<\/p><p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><menclose notation=\"box\"><mstyle displaystyle=\"false\" scriptlevel=\"0\"><\/mstyle><mstyle displaystyle=\"false\" scriptlevel=\"0\"><\/mstyle><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow><mo>\u222b<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mi>cot<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>csc<\/mi><mo>\u2061<\/mo><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><\/mrow><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{\\int \\csc x \\cot x \\, dx = -\\csc x + C}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"boxpad\"><span class=\"mop op-symbol large-op\">\u222b<\/span><span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mop\">cot<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mord mathnormal\">d<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span>\u2212<span class=\"mop\">csc<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">C<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\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>Integration of Cosec x x = 2 \u222bcscx dx Formula \u222b cscx dx = ln|cscx \u2212 cotx| + C Angle (x) 2 rad Result Form ln|csc(2) \u2212 cot(2)| + C Imagine you&#8217;re at an amusement park, riding a roller coaster. As you go up and down, your height keeps changing smoothly\u2014 just like a sine [&hellip;]<\/p>\n","protected":false},"author":116,"featured_media":38585,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[979],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/38566"}],"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=38566"}],"version-history":[{"count":27,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/38566\/revisions"}],"predecessor-version":[{"id":38631,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/38566\/revisions\/38631"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media\/38585"}],"wp:attachment":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media?parent=38566"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/categories?post=38566"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/tags?post=38566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}