{"id":133958,"date":"2023-03-12T17:17:47","date_gmt":"2023-03-12T11:47:47","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=133958"},"modified":"2023-03-12T17:17:47","modified_gmt":"2023-03-12T11:47:47","slug":"evalute-x-2-f0-sin-2x-tan-1-sin-xdx","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/quiz\/evalute-x-2-f0-sin-2x-tan-1-sin-xdx","title":{"rendered":"Evalute x\/2 f0 sin 2x tan-1 (sin x)dx"},"content":{"rendered":"<h2>Question: Evalute x\/2 f0 sin 2x tan-1 (sin x)dx<\/h2>\n<h3>The correct answer is &#8211;<\/h3>\n<div class=\"group w-full text-gray-800 dark:text-gray-100 border-b border-black\/10 dark:border-gray-900\/50 bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0 m-auto\">\n<div class=\"relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<h4>We can start by using integration by parts, with:<\/h4>\n<h4>u = sin(2x) dv = tan\u207b\u00b9(sin x) dx<\/h4>\n<h4>Then:<\/h4>\n<h4>du\/dx = 2cos(2x) v = x\/2 tan\u207b\u00b9(sin x) &#8211; (1\/4) ln(cos x)<\/h4>\n<h4>Using these substitutions, we have:<\/h4>\n<h4>x\/2 f0 sin 2x tan-1(sin x) dx = [x\/2 sin(2x) [x\/2 tan\u207b\u00b9(sin x) &#8211; (1\/4) ln(cos x)]]\u2080\u1d28\/2 &#8211; f0 [x\/2 tan\u207b\u00b9(sin x) &#8211; (1\/4) ln(cos x)] cos(2x) dx<\/h4>\n<h4>Next, we use integration by parts again, with:<\/h4>\n<h4>u = x\/2 tan\u207b\u00b9(sin x) &#8211; (1\/4) ln(cos x) dv = cos(2x) dx<\/h4>\n<h4>Then:<\/h4>\n<h4>du\/dx = (1\/2) [tan\u207b\u00b9(sin x) &#8211; x sin x\/(cos x)\u00b2] + (1\/4) sin x\/cos x v = (1\/2) sin(2x)<\/h4>\n<h4>Substituting these values, we get:<\/h4>\n<h4>x\/2 f0 sin 2x tan-1(sin x) dx = [x\/2 sin(2x) [x\/2 tan\u207b\u00b9(sin x) &#8211; (1\/4) ln(cos x)]]\u2080\u1d28\/2 &#8211; f0 [(1\/2) sin(2x)] [x\/2 tan\u207b\u00b9(sin x) &#8211; (1\/4) ln(cos x)]\u2080\u1d28\/2 &#8211; f0 \u1d28\/2 [1\/2 tan\u207b\u00b9(sin \u1d28\/2) &#8211; (1\/4) ln(cos \u1d28\/2)]<\/h4>\n<h4>Simplifying, we get:<\/h4>\n<h4>x\/2 f0 sin 2x tan-1(sin x) dx = (1\/4) \u1d28 ln 2 &#8211; f0 [(1\/2) sin(2x) tan\u207b\u00b9(sin x) &#8211; (1\/8) ln(cos\u00b2x)]\u2080\u1d28\/2 &#8211; (1\/4) \u1d28 tan\u207b\u00b9(1)<\/h4>\n<h4>Therefore, the answer is (1\/4) \u1d28 ln 2 &#8211; (1\/2) f0 [sin(2x) tan\u207b\u00b9(sin x) &#8211; (1\/4) ln(cos\u00b2x)]\u2080\u1d28\/2 &#8211; (1\/4) \u1d28 tan\u207b\u00b9(1).<\/h4>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"flex justify-between\">\n<div class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-3 md:gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Question: Evalute x\/2 f0 sin 2x tan-1 (sin x)dx The correct answer is &#8211; We can start by using integration by parts, with: u = sin(2x) dv = tan\u207b\u00b9(sin x) dx Then: du\/dx = 2cos(2x) v = x\/2 tan\u207b\u00b9(sin x) &#8211; (1\/4) ln(cos x) Using these substitutions, we have: x\/2 f0 sin 2x tan-1(sin x) [&hellip;]<\/p>\n","protected":false},"author":21842,"featured_media":132233,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12119],"tags":[],"class_list":{"0":"post-133958","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-quiz"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133958","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/users\/21842"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/comments?post=133958"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133958\/revisions"}],"predecessor-version":[{"id":133959,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133958\/revisions\/133959"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media\/132233"}],"wp:attachment":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media?parent=133958"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=133958"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=133958"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}