To evaluate the integral \u222b19yln\u2061(y)\u2009dy\\int_{1}^{9} y \\ln(y) \\, dy\u222b19\u200byln(y)dy, we will use integration by parts<\/strong>.<\/p>\n\n\n\n We will apply the formula for integration by parts:\u222bu\u2009dv=uv\u2212\u222bv\u2009du\\int u \\, dv = uv – \\int v \\, du\u222budv=uv\u2212\u222bvdu<\/p>\n\n\n\n In this case, we choose:<\/p>\n\n\n\n Now, differentiate and integrate:<\/p>\n\n\n\n Using the formula:\u222b19yln\u2061(y)\u2009dy=[y22ln\u2061(y)]19\u2212\u222b19y22\u22c51y\u2009dy\\int_{1}^{9} y \\ln(y) \\, dy = \\left[ \\frac{y^2}{2} \\ln(y) \\right]_{1}^{9} – \\int_{1}^{9} \\frac{y^2}{2} \\cdot \\frac{1}{y} \\, dy\u222b19\u200byln(y)dy=[2y2\u200bln(y)]19\u200b\u2212\u222b19\u200b2y2\u200b\u22c5y1\u200bdy<\/p>\n\n\n\n Simplify the second integral:\u222b19y22\u22c51y\u2009dy=\u222b19y2\u2009dy\\int_{1}^{9} \\frac{y^2}{2} \\cdot \\frac{1}{y} \\, dy = \\int_{1}^{9} \\frac{y}{2} \\, dy\u222b19\u200b2y2\u200b\u22c5y1\u200bdy=\u222b19\u200b2y\u200bdy<\/p>\n\n\n\n Now, evaluate the first term:[y22ln\u2061(y)]19=922ln\u2061(9)\u2212122ln\u2061(1)\\left[ \\frac{y^2}{2} \\ln(y) \\right]_{1}^{9} = \\frac{9^2}{2} \\ln(9) – \\frac{1^2}{2} \\ln(1)[2y2\u200bln(y)]19\u200b=292\u200bln(9)\u2212212\u200bln(1)=812ln\u2061(9)\u22120= \\frac{81}{2} \\ln(9) – 0=281\u200bln(9)\u22120<\/p>\n\n\n\n Since ln\u2061(1)=0\\ln(1) = 0ln(1)=0, this term simplifies to:812ln\u2061(9)\\frac{81}{2} \\ln(9)281\u200bln(9)<\/p>\n\n\n\n Now, evaluate the second integral:\u222b19y2\u2009dy=12\u222b19y\u2009dy=12[y22]19\\int_{1}^{9} \\frac{y}{2} \\, dy = \\frac{1}{2} \\int_{1}^{9} y \\, dy = \\frac{1}{2} \\left[ \\frac{y^2}{2} \\right]_{1}^{9}\u222b19\u200b2y\u200bdy=21\u200b\u222b19\u200bydy=21\u200b[2y2\u200b]19\u200b=12(922\u2212122)=12(812\u221212)= \\frac{1}{2} \\left( \\frac{9^2}{2} – \\frac{1^2}{2} \\right) = \\frac{1}{2} \\left( \\frac{81}{2} – \\frac{1}{2} \\right)=21\u200b(292\u200b\u2212212\u200b)=21\u200b(281\u200b\u221221\u200b)=12\u00d7802=804=20= \\frac{1}{2} \\times \\frac{80}{2} = \\frac{80}{4} = 20=21\u200b\u00d7280\u200b=480\u200b=20<\/p>\n\n\n\n Now combine the two parts:812ln\u2061(9)\u221220\\frac{81}{2} \\ln(9) – 20281\u200bln(9)\u221220<\/p>\n\n\n\n Since ln\u2061(9)=2ln\u2061(3)\\ln(9) = 2 \\ln(3)ln(9)=2ln(3), we can rewrite the result as:812\u00d72ln\u2061(3)\u221220=81ln\u2061(3)\u221220\\frac{81}{2} \\times 2 \\ln(3) – 20 = 81 \\ln(3) – 20281\u200b\u00d72ln(3)\u221220=81ln(3)\u221220<\/p>\n\n\n\n The value of the integral is:81ln\u2061(3)\u22122081 \\ln(3) – 2081ln(3)\u221220<\/p>\n\n\n\n Evaluate the integral. 9 1 y ln(y) dy The Correct Answer and Explanation is: To evaluate the integral \u222b19yln\u2061(y)\u2009dy\\int_{1}^{9} y \\ln(y) \\, dy\u222b19\u200byln(y)dy, we will use integration by parts. Step 1: Identify parts for integration by parts We will apply the formula for integration by parts:\u222bu\u2009dv=uv\u2212\u222bv\u2009du\\int u \\, dv = uv – \\int v \\, […]<\/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-247548","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/247548","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=247548"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/247548\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=247548"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=247548"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=247548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Identify parts for integration by parts<\/h3>\n\n\n\n
\n
\n
Step 2: Apply the integration by parts formula<\/h3>\n\n\n\n
Step 3: Compute the first part<\/h3>\n\n\n\n
Step 4: Compute the second part<\/h3>\n\n\n\n
Step 5: Combine the results<\/h3>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
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