To compute the definite integral:<\/p>\n\n\n\n
I=\u222bee3(ln\u2061x)2x\u2009dxI = \\int_{e}^{e^3} \\frac{(\\ln x)^2}{x} \\,dx<\/p>\n\n\n\n
we use the substitution method. Let u=ln\u2061xu = \\ln x, then du=dxxdu = \\frac{dx}{x}, transforming the integral into:<\/p>\n\n\n\n
I=\u222bln\u2061eln\u2061e3u2\u2009duI = \\int_{\\ln e}^{\\ln e^3} u^2 \\, du<\/p>\n\n\n\n
Since ln\u2061e=1\\ln e = 1 and ln\u2061e3=3\\ln e^3 = 3, the limits of integration adjust accordingly:<\/p>\n\n\n\n
I=\u222b13u2\u2009duI = \\int_{1}^{3} u^2 \\,du<\/p>\n\n\n\n
Now, integrating u2u^2:<\/p>\n\n\n\n
\u222bu2\u2009du=u33\\int u^2 \\,du = \\frac{u^3}{3}<\/p>\n\n\n\n
Evaluating this expression from u=1u = 1 to u=3u = 3:<\/p>\n\n\n\n
I=[u33]13I = \\left[ \\frac{u^3}{3} \\right]_{1}^{3}<\/p>\n\n\n\n
I=333\u2212133I = \\frac{3^3}{3} – \\frac{1^3}{3}<\/p>\n\n\n\n
I=273\u221213I = \\frac{27}{3} – \\frac{1}{3}<\/p>\n\n\n\n
I=9\u221213I = 9 – \\frac{1}{3}<\/p>\n\n\n\n
I=823I = 8 \\frac{2}{3}<\/p>\n\n\n\n
The exact value of the definite integral is 263\\frac{26}{3} or 8 <\/strong>23\\frac{2}{3}.<\/p>\n\n\n\n The integration process involves recognizing that the given integral can be simplified using substitution. By letting u=ln\u2061xu = \\ln x, we effectively transform the complicated logarithmic expression into a polynomial integral. This technique streamlines the computation by reducing the complexity associated with logarithmic functions.<\/p>\n\n\n\n After substituting u=ln\u2061xu = \\ln x and converting dx\/xdx\/x into dudu, the integral simplifies to \u222bu2\u2009du\\int u^2 \\,du. This transformation enables straightforward polynomial integration. The antiderivative of u2u^2 is determined by the power rule, giving u3\/3u^3\/3, which is then evaluated at the limits u=1u = 1 and u=3u = 3.<\/p>\n\n\n\n Substituting these values into the antiderivative results in (27\/3\u22121\/3)(27\/3 – 1\/3), which simplifies to 26\/326\/3 or 8238 \\frac{2}{3}. Thus, the definite integral is computed precisely, ensuring accuracy through each step of substitution, integration, and evaluation.<\/p>\n\n\n\n This method is particularly effective for logarithmic integrals where exponentiation is involved, demonstrating the utility of substitution when tackling problems that contain expressions of ln\u2061x\\ln x.<\/p>\n\n\n\n Compute the definite integral lnx)2 dx Give the exact value. The Correct Answer and Explanation is: To compute the definite integral: I=\u222bee3(ln\u2061x)2x\u2009dxI = \\int_{e}^{e^3} \\frac{(\\ln x)^2}{x} \\,dx we use the substitution method. Let u=ln\u2061xu = \\ln x, then du=dxxdu = \\frac{dx}{x}, transforming the integral into: I=\u222bln\u2061eln\u2061e3u2\u2009duI = \\int_{\\ln e}^{\\ln e^3} u^2 \\, du Since ln\u2061e=1\\ln […]<\/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-237137","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237137","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=237137"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237137\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=237137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=237137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=237137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Explanation<\/h3>\n\n\n\n
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<\/figure>\n","protected":false},"excerpt":{"rendered":"