{"id":247552,"date":"2025-07-07T18:15:23","date_gmt":"2025-07-07T18:15:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=247552"},"modified":"2025-07-07T18:15:25","modified_gmt":"2025-07-07T18:15:25","slug":"find-the-integral-of-5y-ln-y-dy-sy-ln-y-dy-c","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/07\/find-the-integral-of-5y-ln-y-dy-sy-ln-y-dy-c\/","title":{"rendered":"Find the integral of 5y ln y dy. | s(y ln y) dy +C"},"content":{"rendered":"\n<p>Find the integral of 5y ln y dy. | s(y ln y) dy +C<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To evaluate the integral of 5yln\u2061y\u2009dy5y \\ln y \\, dy5ylnydy, we will use <strong>integration by parts<\/strong>, which is based on the formula:\u222bu\u2009dv=uv\u2212\u222bv\u2009du\\int u \\, dv = uv &#8211; \\int v \\, du\u222budv=uv\u2212\u222bvdu<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Set up uuu and dvdvdv<\/h3>\n\n\n\n<p>We choose u=ln\u2061yu = \\ln yu=lny, because its derivative simplifies the expression, and dv=5y\u2009dydv = 5y \\, dydv=5ydy, because the integral of yyy is straightforward.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>u=ln\u2061yu = \\ln yu=lny so that du=1y\u2009dydu = \\frac{1}{y} \\, dydu=y1\u200bdy<\/li>\n\n\n\n<li>dv=5y\u2009dydv = 5y \\, dydv=5ydy so that v=5y22v = \\frac{5y^2}{2}v=25y2\u200b<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Apply the Integration by Parts Formula<\/h3>\n\n\n\n<p>Substitute into the formula for integration by parts:\u222b5yln\u2061y\u2009dy=5y22ln\u2061y\u2212\u222b5y22\u22c51y\u2009dy\\int 5y \\ln y \\, dy = \\frac{5y^2}{2} \\ln y &#8211; \\int \\frac{5y^2}{2} \\cdot \\frac{1}{y} \\, dy\u222b5ylnydy=25y2\u200blny\u2212\u222b25y2\u200b\u22c5y1\u200bdy<\/p>\n\n\n\n<p>Simplifying the second integral:\u222b5y22\u22c51y\u2009dy=\u222b5y2\u2009dy\\int \\frac{5y^2}{2} \\cdot \\frac{1}{y} \\, dy = \\int \\frac{5y}{2} \\, dy\u222b25y2\u200b\u22c5y1\u200bdy=\u222b25y\u200bdy<\/p>\n\n\n\n<p>Now integrate 5y2\\frac{5y}{2}25y\u200b:\u222b5y2\u2009dy=5y24\\int \\frac{5y}{2} \\, dy = \\frac{5y^2}{4}\u222b25y\u200bdy=45y2\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Put Everything Together<\/h3>\n\n\n\n<p>Now, substitute this back into the original expression:\u222b5yln\u2061y\u2009dy=5y22ln\u2061y\u22125y24+C\\int 5y \\ln y \\, dy = \\frac{5y^2}{2} \\ln y &#8211; \\frac{5y^2}{4} + C\u222b5ylnydy=25y2\u200blny\u221245y2\u200b+C<\/p>\n\n\n\n<p>This is the final result for the integral of 5yln\u2061y\u2009dy5y \\ln y \\, dy5ylnydy.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>The integral of 5yln\u2061y\u2009dy5y \\ln y \\, dy5ylnydy is:5y22ln\u2061y\u22125y24+C\\frac{5y^2}{2} \\ln y &#8211; \\frac{5y^2}{4} + C25y2\u200blny\u221245y2\u200b+C<\/p>\n\n\n\n<p>where CCC is the constant of integration. This solution combines integration by parts and simplifies the terms to give the final expression.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-1027.jpeg\" alt=\"\" class=\"wp-image-247553\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the integral of 5y ln y dy. | s(y ln y) dy +C The Correct Answer and Explanation is: To evaluate the integral of 5yln\u2061y\u2009dy5y \\ln y \\, dy5ylnydy, we will use integration by parts, which is based on the formula:\u222bu\u2009dv=uv\u2212\u222bv\u2009du\\int u \\, dv = uv &#8211; \\int v \\, du\u222budv=uv\u2212\u222bvdu Step 1: Set [&hellip;]<\/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-247552","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/247552","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=247552"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/247552\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=247552"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=247552"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=247552"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}