{"id":133956,"date":"2023-03-12T17:14:19","date_gmt":"2023-03-12T11:44:19","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=133956"},"modified":"2023-03-12T17:14:19","modified_gmt":"2023-03-12T11:44:19","slug":"find-x4-x-1x21dx","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/quiz\/find-x4-x-1x21dx","title":{"rendered":"find x4\/(x-1)(x2+1)dx"},"content":{"rendered":"<h2>Question: find x4\/(x-1)(x2+1)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 use partial fraction decomposition to solve the integral:<\/h4>\n<h4>x^4 \/ [(x-1)(x^2+1)] dx<\/h4>\n<h4>First, we factor the denominator:<\/h4>\n<h4>x^4 \/ [(x-1)(x^2+1)] = A\/(x-1) + Bx\/(x^2+1)<\/h4>\n<h4>Multiplying both sides by the denominator and simplifying, we get:<\/h4>\n<h4>x^4 = A(x^2+1) + Bx(x-1)<\/h4>\n<h4>Now, we substitute some values to find the values of A and B:<\/h4>\n<h4>If we let x=1, then we get:<\/h4>\n<h4>1^4 = A(1^2+1) + B(1)(1-1) = 2A<\/h4>\n<h4>A = 1\/2<\/h4>\n<h4>If we let x=0, then we get:<\/h4>\n<h4>0^4 = A(0^2+1) + B(0)(0-1) = A<\/h4>\n<h4>A = 0<\/h4>\n<h4>So, we have:<\/h4>\n<h4>Bx(x-1) = x^4 &#8211; A(x^2+1)<\/h4>\n<h4>Substituting the values of A and B, we get:<\/h4>\n<h4>Bx(x-1) = x^4 &#8211; (1\/2)(x^2+1)<\/h4>\n<h4>Now, we can integrate each term separately:<\/h4>\n<h4>\u222b[x^4 \/ (x-1)(x^2+1)] dx = \u222b[1\/2(x-1) + (1\/2)x &#8211; (1\/2)(x\/(x^2+1))] dx<\/h4>\n<h4>= 1\/2 ln|x-1| + 1\/4 ln(x^2+1) &#8211; 1\/2 ln|x^2+1| + C<\/h4>\n<h4>where C is the constant of integration.<\/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: find x4\/(x-1)(x2+1)dx The correct answer is &#8211; We can use partial fraction decomposition to solve the integral: x^4 \/ [(x-1)(x^2+1)] dx First, we factor the denominator: x^4 \/ [(x-1)(x^2+1)] = A\/(x-1) + Bx\/(x^2+1) Multiplying both sides by the denominator and simplifying, we get: x^4 = A(x^2+1) + Bx(x-1) Now, we substitute some values to [&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-133956","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\/133956","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=133956"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133956\/revisions"}],"predecessor-version":[{"id":133957,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133956\/revisions\/133957"}],"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=133956"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=133956"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=133956"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}