To solve the indefinite integral of the expression:\u222bcsc\u2061(x)cot\u2061(x)\u22c516csc\u20612(x)\u2009dx\\int \\csc(x) \\cot(x) \\cdot \\sqrt{16 \\csc^2(x)} \\, dx\u222bcsc(x)cot(x)\u22c516csc2(x)\u200bdx<\/p>\n\n\n\n
Step 1: Simplify the expression<\/h3>\n\n\n\n
We start by simplifying the square root term:16csc\u20612(x)=4csc\u2061(x)\\sqrt{16 \\csc^2(x)} = 4 \\csc(x)16csc2(x)\u200b=4csc(x)<\/p>\n\n\n\n
Thus, the integral becomes:\u222bcsc\u2061(x)cot\u2061(x)\u22c54csc\u2061(x)\u2009dx\\int \\csc(x) \\cot(x) \\cdot 4 \\csc(x) \\, dx\u222bcsc(x)cot(x)\u22c54csc(x)dx<\/p>\n\n\n\n
This simplifies further to:\u222b4csc\u20612(x)cot\u2061(x)\u2009dx\\int 4 \\csc^2(x) \\cot(x) \\, dx\u222b4csc2(x)cot(x)dx<\/p>\n\n\n\n
Step 2: Make a substitution<\/h3>\n\n\n\n
Notice that the integral contains both csc\u20612(x)\\csc^2(x)csc2(x) and cot\u2061(x)\\cot(x)cot(x), which suggests using substitution. We will use the substitution:u=cot\u2061(x)u = \\cot(x)u=cot(x)<\/p>\n\n\n\n
Thus, the derivative of cot\u2061(x)\\cot(x)cot(x) with respect to xxx is:dudx=\u2212csc\u20612(x)\\frac{du}{dx} = -\\csc^2(x)dxdu\u200b=\u2212csc2(x)<\/p>\n\n\n\n
or equivalently:du=\u2212csc\u20612(x)\u2009dxdu = -\\csc^2(x) \\, dxdu=\u2212csc2(x)dx<\/p>\n\n\n\n
Step 3: Substitute and integrate<\/h3>\n\n\n\n
Substitute u=cot\u2061(x)u = \\cot(x)u=cot(x) and du=\u2212csc\u20612(x)\u2009dxdu = -\\csc^2(x) \\, dxdu=\u2212csc2(x)dx into the integral:\u222b4csc\u20612(x)cot\u2061(x)\u2009dx=\u22124\u222bu\u2009du\\int 4 \\csc^2(x) \\cot(x) \\, dx = -4 \\int u \\, du\u222b4csc2(x)cot(x)dx=\u22124\u222budu<\/p>\n\n\n\n
Now, we can integrate with respect to uuu:\u22124\u222bu\u2009du=\u22124\u22c5u22=\u22122u2-4 \\int u \\, du = -4 \\cdot \\frac{u^2}{2} = -2u^2\u22124\u222budu=\u22124\u22c52u2\u200b=\u22122u2<\/p>\n\n\n\n
Step 4: Substitute back<\/h3>\n\n\n\n
Now, we substitute u=cot\u2061(x)u = \\cot(x)u=cot(x) back into the expression:\u22122u2=\u22122cot\u20612(x)-2u^2 = -2 \\cot^2(x)\u22122u2=\u22122cot2(x)<\/p>\n\n\n\n
Step 5: Add the constant of integration<\/h3>\n\n\n\n
Finally, since we are evaluating an indefinite integral, we add the constant of integration CCC:\u222bcsc\u2061(x)cot\u2061(x)\u22c516csc\u20612(x)\u2009dx=\u22122cot\u20612(x)+C\\int \\csc(x) \\cot(x) \\cdot \\sqrt{16 \\csc^2(x)} \\, dx = -2 \\cot^2(x) + C\u222bcsc(x)cot(x)\u22c516csc2(x)\u200bdx=\u22122cot2(x)+C<\/p>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
\u22122cot\u20612(x)+C\\boxed{-2 \\cot^2(x) + C}\u22122cot2(x)+C\u200b<\/p>\n\n\n\n
Explanation:<\/h3>\n\n\n\n
In this solution, we used substitution to simplify the integral. By recognizing the relationship between the trigonometric functions and using the derivative of cot\u2061(x)\\cot(x)cot(x), we were able to transform the integral into a more straightforward form, which was then solved using basic integration rules. The constant of integration CCC accounts for the fact that the integral represents a family of functions, all differing by a constant.<\/p>\n\n\n\n Find the indefinite integral. (Use C for the constant of integration.) csc(x) cot(x) V 16 csc2 X The Correct Answer and Explanation is: To solve the indefinite integral of the expression:\u222bcsc\u2061(x)cot\u2061(x)\u22c516csc\u20612(x)\u2009dx\\int \\csc(x) \\cot(x) \\cdot \\sqrt{16 \\csc^2(x)} \\, dx\u222bcsc(x)cot(x)\u22c516csc2(x)\u200bdx Step 1: Simplify the expression We start by simplifying the square root term:16csc\u20612(x)=4csc\u2061(x)\\sqrt{16 \\csc^2(x)} = 4 \\csc(x)16csc2(x)\u200b=4csc(x) […]<\/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-245838","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245838","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=245838"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245838\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-729.jpeg\")
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