{"id":407,"date":"2025-02-13T19:44:40","date_gmt":"2025-02-13T19:44:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-using-substitution-fresh-take\/"},"modified":"2025-02-13T19:44:40","modified_gmt":"2025-02-13T19:44:40","slug":"integration-using-substitution-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-using-substitution-fresh-take\/","title":{"raw":"Integration using Substitution: Fresh Take","rendered":"Integration using Substitution: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Identify when to use substitution to simplify and solve integrals<\/li>\n\t<li>Apply substitution methods to find indefinite integrals<\/li>\n\t<li>Apply substitution methods to find definite integrals<\/li>\n<\/ul>\n<\/section>\n<h2>Substitution for Indefinite Integrals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Integration by substitution is a technique for evaluating integrals. It's useful when the integrand is the result of a chain-rule derivative<\/li>\n\t<li class=\"whitespace-normal break-words\">The method involves changing variables to simplify the integral<\/li>\n\t<li class=\"whitespace-normal break-words\">Key form to recognize:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\int f(g(x))g'(x)dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitution Theorem:\n\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">For [latex]u = g(x)[\/latex] where [latex]g'(x)[\/latex] is continuous: [latex]\\int f(g(x))g'(x)dx = \\int f(u)du = F(u) + C = F(g(x)) + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Process:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Choose [latex]u = g(x)[\/latex] such that [latex]g'(x)[\/latex] is part of the integrand<\/li>\n\t<li class=\"whitespace-normal break-words\">Express the integral in terms of [latex]u[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Integrate with respect to [latex]u[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitute back to express the result in terms of [latex]x[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Sometimes need to adjust by a constant factor when [latex]du[\/latex] doesn't match exactly<\/li>\n\t<li class=\"whitespace-normal break-words\">May need to express [latex]x[\/latex] in terms of [latex]u[\/latex] to eliminate all [latex]x[\/latex] terms<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int 3{x}^{2}{({x}^{3}-3)}^{2}dx.[\/latex]<\/p>\n<div id=\"fs-id1170572554378\" class=\"exercise\">\n<div id=\"fs-id1170571419835\" class=\"commentary\">\n<p>[reveal-answer q=\"43088721\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"43088721\"]<\/p>\n<p id=\"fs-id1170572554429\">Let [latex]u={x}^{3}-3.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"fs-id1170572628419\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572628419\"]<\/p>\n<p id=\"fs-id1170572628419\">[latex]\\displaystyle\\int 3{x}^{2}{({x}^{3}-3)}^{2}dx=\\frac{1}{3}{({x}^{3}-3)}^{3}+C[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int {x}^{2}{({x}^{3}+5)}^{9}dx.[\/latex]<\/p>\n<p>[reveal-answer q=\"9013327\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"9013327\"]<\/p>\n<p>Multiply the <em>du<\/em> equation by [latex]\\frac{1}{3}.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"fs-id1170572628420\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572628420\"]<\/p>\n<div id=\"qfs-id1170573361491\" class=\"hidden-answer\">\n<p id=\"fs-id1170573361491\">[latex]\\frac{{({x}^{3}+5)}^{10}}{30}+C[\/latex]<\/p>\n<\/div>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Use substitution to evaluate the integral [latex]\\displaystyle\\int \\frac{ \\cos t}{{ \\sin }^{2}t}dt.[\/latex]<\/p>\n<p>[reveal-answer q=\"fs-id1170572628519\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572628519\"]<\/p>\n<div id=\"qfs-id1170571171177\" class=\"hidden-answer\">\n<p id=\"fs-id1170571171177\">[latex]-\\dfrac{1}{ \\sin t}+C[\/latex]<\/p>\n<\/div>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571087102\">Use substitution to evaluate the indefinite integral [latex]\\displaystyle\\int { \\cos }^{3}t \\sin tdt.[\/latex]<\/p>\n<p>[reveal-answer q=\"fs-id1170572628520\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572628520\"]<\/p>\n<div id=\"qfs-id1170571285848\" class=\"hidden-answer\">\n<p id=\"fs-id1170571285848\">[latex]-\\dfrac{{ \\cos }^{4}t}{4}+C[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/section>\n<h2>Substitution for Definite Integrals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Substitution can be applied to definite integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">The limits of integration must be adjusted when changing variables<\/li>\n\t<li class=\"whitespace-normal break-words\">The technique combines substitution with the Fundamental Theorem of Calculus<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitution Theorem for Definite Integrals:\n\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">If [latex]u = g(x)[\/latex] and [latex]g'(x)[\/latex] is continuous on [latex][a,b][\/latex], then: [latex]\\int_a^b f(g(x))g'(x)dx = \\int_{g(a)}^{g(b)} f(u)du[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Process:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Choose [latex]u = g(x)[\/latex] as in indefinite integration<\/li>\n\t<li class=\"whitespace-normal break-words\">Express the integrand in terms of [latex]u[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Change the limits of integration from [latex]x[\/latex] to [latex]u[\/latex] values<\/li>\n\t<li class=\"whitespace-normal break-words\">Integrate with respect to [latex]u[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Evaluate the integral using the new limits<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Alternative Approach:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Perform substitution without changing limits<\/li>\n\t<li class=\"whitespace-normal break-words\">Find the antiderivative in terms of [latex]u[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitute back to [latex]x[\/latex] before evaluating at the original limits<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Combining Techniques:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Substitution may be used alongside other integration techniques<\/li>\n\t<li class=\"whitespace-normal break-words\">Trigonometric identities might be needed before substitution<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use substitution to evaluate the definite integral [latex]{\\displaystyle\\int }_{-1}^{0}y{(2{y}^{2}-3)}^{5}dy.[\/latex]<\/p>\n<p>[reveal-answer q=\"fs-id1170572628523\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572628523\"]<\/p>\n<div id=\"qfs-id1170571285848\" class=\"hidden-answer\">\n<p id=\"fs-id1170571285848\">[latex]\\frac{91}{3}[\/latex]<\/p>\n<\/div>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Use substitution to evaluate [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2} \\cos \\left(\\frac{\\pi }{2}{x}^{3}\\right)dx.[\/latex]<\/p>\n<p>[reveal-answer q=\"fs-id1170572587526\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572587526\"]<\/p>\n<div id=\"qfs-id1170573438316\" class=\"hidden-answer\">\n<p id=\"fs-id1170573438316\">[latex]\\frac{2}{3\\pi }\\approx 0.2122[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170571099763\" class=\"exercise\">\n<p>[\/hidden-answer]<\/p>\n<\/div>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify when to use substitution to simplify and solve integrals<\/li>\n<li>Apply substitution methods to find indefinite integrals<\/li>\n<li>Apply substitution methods to find definite integrals<\/li>\n<\/ul>\n<\/section>\n<h2>Substitution for Indefinite Integrals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Integration by substitution is a technique for evaluating integrals. It&#8217;s useful when the integrand is the result of a chain-rule derivative<\/li>\n<li class=\"whitespace-normal break-words\">The method involves changing variables to simplify the integral<\/li>\n<li class=\"whitespace-normal break-words\">Key form to recognize:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int f(g(x))g'(x)dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitution Theorem:\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]u = g(x)[\/latex] where [latex]g'(x)[\/latex] is continuous: [latex]\\int f(g(x))g'(x)dx = \\int f(u)du = F(u) + C = F(g(x)) + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose [latex]u = g(x)[\/latex] such that [latex]g'(x)[\/latex] is part of the integrand<\/li>\n<li class=\"whitespace-normal break-words\">Express the integral in terms of [latex]u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Integrate with respect to [latex]u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute back to express the result in terms of [latex]x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Sometimes need to adjust by a constant factor when [latex]du[\/latex] doesn&#8217;t match exactly<\/li>\n<li class=\"whitespace-normal break-words\">May need to express [latex]x[\/latex] in terms of [latex]u[\/latex] to eliminate all [latex]x[\/latex] terms<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int 3{x}^{2}{({x}^{3}-3)}^{2}dx.[\/latex]<\/p>\n<div id=\"fs-id1170572554378\" class=\"exercise\">\n<div id=\"fs-id1170571419835\" class=\"commentary\">\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q43088721\">Hint<\/button><\/p>\n<div id=\"q43088721\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572554429\">Let [latex]u={x}^{3}-3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572628419\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572628419\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572628419\">[latex]\\displaystyle\\int 3{x}^{2}{({x}^{3}-3)}^{2}dx=\\frac{1}{3}{({x}^{3}-3)}^{3}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int {x}^{2}{({x}^{3}+5)}^{9}dx.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q9013327\">Hint<\/button><\/p>\n<div id=\"q9013327\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply the <em>du<\/em> equation by [latex]\\frac{1}{3}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572628420\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572628420\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"qfs-id1170573361491\" class=\"hidden-answer\">\n<p id=\"fs-id1170573361491\">[latex]\\frac{{({x}^{3}+5)}^{10}}{30}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use substitution to evaluate the integral [latex]\\displaystyle\\int \\frac{ \\cos t}{{ \\sin }^{2}t}dt.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572628519\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572628519\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"qfs-id1170571171177\" class=\"hidden-answer\">\n<p id=\"fs-id1170571171177\">[latex]-\\dfrac{1}{ \\sin t}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571087102\">Use substitution to evaluate the indefinite integral [latex]\\displaystyle\\int { \\cos }^{3}t \\sin tdt.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572628520\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572628520\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"qfs-id1170571285848\" class=\"hidden-answer\">\n<p id=\"fs-id1170571285848\">[latex]-\\dfrac{{ \\cos }^{4}t}{4}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Substitution for Definite Integrals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitution can be applied to definite integrals<\/li>\n<li class=\"whitespace-normal break-words\">The limits of integration must be adjusted when changing variables<\/li>\n<li class=\"whitespace-normal break-words\">The technique combines substitution with the Fundamental Theorem of Calculus<\/li>\n<li class=\"whitespace-normal break-words\">Substitution Theorem for Definite Integrals:\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]u = g(x)[\/latex] and [latex]g'(x)[\/latex] is continuous on [latex][a,b][\/latex], then: [latex]\\int_a^b f(g(x))g'(x)dx = \\int_{g(a)}^{g(b)} f(u)du[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Choose [latex]u = g(x)[\/latex] as in indefinite integration<\/li>\n<li class=\"whitespace-normal break-words\">Express the integrand in terms of [latex]u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Change the limits of integration from [latex]x[\/latex] to [latex]u[\/latex] values<\/li>\n<li class=\"whitespace-normal break-words\">Integrate with respect to [latex]u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Evaluate the integral using the new limits<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Alternative Approach:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Perform substitution without changing limits<\/li>\n<li class=\"whitespace-normal break-words\">Find the antiderivative in terms of [latex]u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute back to [latex]x[\/latex] before evaluating at the original limits<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Combining Techniques:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitution may be used alongside other integration techniques<\/li>\n<li class=\"whitespace-normal break-words\">Trigonometric identities might be needed before substitution<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Use substitution to evaluate the definite integral [latex]{\\displaystyle\\int }_{-1}^{0}y{(2{y}^{2}-3)}^{5}dy.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572628523\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572628523\" class=\"hidden-answer\" style=\"display: none\">\n<div class=\"hidden-answer\">\n<p>[latex]\\frac{91}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use substitution to evaluate [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2} \\cos \\left(\\frac{\\pi }{2}{x}^{3}\\right)dx.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572587526\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572587526\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"qfs-id1170573438316\" class=\"hidden-answer\">\n<p id=\"fs-id1170573438316\">[latex]\\frac{2}{3\\pi }\\approx 0.2122[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170571099763\" class=\"exercise\">\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":399,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/407"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/407\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/399"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/407\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=407"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=407"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=407"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}