![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/image-361.png\")
<\/figure>\n\n\n\nProblem:<\/h3>\n\n\n\n
Evaluate the integral:\u222bcot\u2061(3x)\u2009dx\\int \\cot(3x)\\,dx\u222bcot(3x)dx<\/p>\n\n\n\n
\n\n\n\n
Correct Answer:<\/h3>\n\n\n\n
\u222bcot\u2061(3x)\u2009dx=13ln\u2061\u2223sin\u2061(3x)\u2223+C\\int \\cot(3x)\\,dx = \\frac{1}{3} \\ln\\left|\\sin(3x)\\right| + C\u222bcot(3x)dx=31\u200bln\u2223sin(3x)\u2223+C<\/p>\n\n\n\n
\n\n\n\n
Explanation <\/h3>\n\n\n\n
To evaluate the integral \u222bcot\u2061(3x)\u2009dx\\int \\cot(3x)\\,dx\u222bcot(3x)dx, we begin by recalling the identity:cot\u2061(u)=cos\u2061(u)sin\u2061(u)\\cot(u) = \\frac{\\cos(u)}{\\sin(u)}cot(u)=sin(u)cos(u)\u200b<\/p>\n\n\n\n
So the integral becomes:\u222bcot\u2061(3x)\u2009dx=\u222bcos\u2061(3x)sin\u2061(3x)\u2009dx\\int \\cot(3x)\\,dx = \\int \\frac{\\cos(3x)}{\\sin(3x)}\\,dx\u222bcot(3x)dx=\u222bsin(3x)cos(3x)\u200bdx<\/p>\n\n\n\n
This form suggests using substitution<\/strong>. Let:u=sin\u2061(3x)u = \\sin(3x)u=sin(3x)<\/p>\n\n\n\n Then, compute the derivative of uuu with respect to xxx:dudx=3cos\u2061(3x)\u21d2du=3cos\u2061(3x)\u2009dx\\frac{du}{dx} = 3\\cos(3x) \\Rightarrow du = 3\\cos(3x)\\,dxdxdu\u200b=3cos(3x)\u21d2du=3cos(3x)dx<\/p>\n\n\n\n Solving for dxdxdx:cos\u2061(3x)\u2009dx=13\u2009du\\cos(3x)\\,dx = \\frac{1}{3}\\,ducos(3x)dx=31\u200bdu<\/p>\n\n\n\n Now, return to the integral and substitute:\u222bcos\u2061(3x)sin\u2061(3x)\u2009dx=\u222b1u\u22c513\u2009du=13\u222b1u\u2009du\\int \\frac{\\cos(3x)}{\\sin(3x)}\\,dx = \\int \\frac{1}{u} \\cdot \\frac{1}{3}\\,du = \\frac{1}{3} \\int \\frac{1}{u}\\,du\u222bsin(3x)cos(3x)\u200bdx=\u222bu1\u200b\u22c531\u200bdu=31\u200b\u222bu1\u200bdu<\/p>\n\n\n\n This is a standard integral:13ln\u2061\u2223u\u2223+C\\frac{1}{3} \\ln|u| + C31\u200bln\u2223u\u2223+C<\/p>\n\n\n\n Now substitute back for u=sin\u2061(3x)u = \\sin(3x)u=sin(3x):13ln\u2061\u2223sin\u2061(3x)\u2223+C\\frac{1}{3} \\ln|\\sin(3x)| + C31\u200bln\u2223sin(3x)\u2223+C<\/p>\n\n\n\n Thus, the final answer is:13ln\u2061\u2223sin\u2061(3x)\u2223+C\\boxed{\\frac{1}{3} \\ln\\left|\\sin(3x)\\right| + C}31\u200bln\u2223sin(3x)\u2223+C\u200b<\/p>\n\n\n\n This result is correct and reflects both the chain rule and substitution method. The provided screenshot shows incorrect substitution steps for u=sin\u2061(x)u = \\sin(x)u=sin(x) and du=1x\u2009dxdu = \\frac{1}{x} \\, dxdu=x1\u200bdx, which don\u2019t match the given integrand. However, the final answer box is correct<\/strong>, so the evaluation was done properly despite the substitution error in the earlier step.<\/p>\n\n\n\n Consider the following integral.Find a substitution to rewrite the integrand as . Evaluate the given integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.) Problem: Evaluate the integral:\u222bcot\u2061(3x)\u2009dx\\int \\cot(3x)\\,dx\u222bcot(3x)dx Correct Answer: \u222bcot\u2061(3x)\u2009dx=13ln\u2061\u2223sin\u2061(3x)\u2223+C\\int \\cot(3x)\\,dx = \\frac{1}{3} \\ln\\left|\\sin(3x)\\right| + C\u222bcot(3x)dx=31\u200bln\u2223sin(3x)\u2223+C Explanation To evaluate the integral \u222bcot\u2061(3x)\u2009dx\\int \\cot(3x)\\,dx\u222bcot(3x)dx, we begin by recalling […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-231619","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231619","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=231619"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/231619\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=231619"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=231619"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=231619"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner4-931.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"