{"id":415,"date":"2025-02-13T19:44:44","date_gmt":"2025-02-13T19:44:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integrals-resulting-in-inverse-trigonometric-functions-fresh-take\/"},"modified":"2025-02-13T19:44:44","modified_gmt":"2025-02-13T19:44:44","slug":"integrals-resulting-in-inverse-trigonometric-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integrals-resulting-in-inverse-trigonometric-functions-fresh-take\/","title":{"raw":"Integrals Resulting in Inverse Trigonometric Functions: Fresh Take","rendered":"Integrals Resulting in Inverse Trigonometric Functions: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Calculate integrals that lead to inverse trigonometric function solutions<\/li>\n<\/ul>\n<\/section>\n<h2>Integrals Resulting in Inverse Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Integration formulas yielding inverse trigonometric functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Domain restrictions for inverse trigonometric functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Connection between derivatives of inverse trig functions and these integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">Application of these formulas to various types of integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">Handling negative integrands in inverse trig integrals<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Key Formulas<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{du}{\\sqrt{a^2 - u^2}} = \\sin^{-1}(\\frac{u}{|a|}) + C[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{du}{a^2 + u^2} = \\frac{1}{a}\\tan^{-1}(\\frac{u}{a}) + C[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{du}{u\\sqrt{u^2 - a^2}} = \\frac{1}{|a|}\\sec^{-1}(\\frac{|u|}{a}) + C[\/latex]<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Key Concepts<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Domain Restrictions:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Inverse trig functions have restricted domains<\/li>\n\t<li class=\"whitespace-normal break-words\">Solutions must respect these domain restrictions<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Recognizing Integrands:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Identify integrands that match standard forms<\/li>\n\t<li class=\"whitespace-normal break-words\">Use substitution to transform integrals into these forms<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Relationship to Derivatives:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">These integrals are related to derivatives of inverse trig functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Understanding this connection aids in application<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Handling Negative Integrands:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Factor out [latex]-1[\/latex] from negative integrands<\/li>\n\t<li class=\"whitespace-normal break-words\">Use existing formulas rather than memorizing new ones<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitution Techniques:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Often necessary to transform integrals into standard forms<\/li>\n\t<li class=\"whitespace-normal break-words\">May involve adjusting constants or variables<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1-16{x}^{2}}}.[\/latex]<\/p>\n\n[reveal-answer q=\"4782055\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"4782055\"]\n\n<p id=\"fs-id1170572089952\">Substitute [latex]u=4x[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"fs-id1170572480280\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572480280\"]<\/p>\n<p>[latex]\\frac{1}{4}\\phantom{\\rule{0.05em}{0ex}}{ \\sin }^{-1}(4x)+C[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/yVwFRhfDP_Y?controls=0&amp;start=173&amp;end=278&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.7IntegralsResultingInInverseTrigonometricFunctions173to278_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.7 Integrals Resulting in Inverse Trigonometric Functions\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the indefinite integral using an inverse trigonometric function and substitution for [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{9-{x}^{2}}}.[\/latex]<\/p>\n<p id=\"fs-id1170572557808\">[reveal-answer q=\"fs-id1170572557808\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572557808\"][latex]{ \\sin }^{-1}\\left(\\frac{x}{3}\\right)+C[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/yVwFRhfDP_Y?controls=0&amp;start=365&amp;end=404&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.7IntegralsResultingInInverseTrigonometricFunctions365to404_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.7 Integrals Resulting in Inverse Trigonometric Functions\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int \\frac{dx}{25+4{x}^{2}}.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1170571562665\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170571562665\"]\n\n<p id=\"fs-id1170571562665\">[latex]\\frac{1}{10}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}\\left(\\frac{2x}{5}\\right)+C[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]\\displaystyle\\int \\frac{dx}{16+{x}^{2}}.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1170571609481\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170571609481\"]\n\n<p id=\"fs-id1170571609481\">[latex]\\frac{1}{4}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{x}{4})+C[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572369374\">Evaluate the definite integral [latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{dx}{4+{x}^{2}}.[\/latex]<\/p>\n<p>[reveal-answer q=\"fs-id1170572176762\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572176762\"]<\/p>\n<p id=\"fs-id1170572176762\">[latex]\\frac{\\pi }{8}[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/yVwFRhfDP_Y?controls=0&amp;start=973&amp;end=1054&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.7IntegralsResultingInInverseTrigonometricFunctions973to1054_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.7 Integrals Resulting in Inverse Trigonometric Functions\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate integrals that lead to inverse trigonometric function solutions<\/li>\n<\/ul>\n<\/section>\n<h2>Integrals Resulting in Inverse Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Integration formulas yielding inverse trigonometric functions<\/li>\n<li class=\"whitespace-normal break-words\">Domain restrictions for inverse trigonometric functions<\/li>\n<li class=\"whitespace-normal break-words\">Connection between derivatives of inverse trig functions and these integrals<\/li>\n<li class=\"whitespace-normal break-words\">Application of these formulas to various types of integrals<\/li>\n<li class=\"whitespace-normal break-words\">Handling negative integrands in inverse trig integrals<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Key Formulas<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{du}{\\sqrt{a^2 - u^2}} = \\sin^{-1}(\\frac{u}{|a|}) + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{du}{a^2 + u^2} = \\frac{1}{a}\\tan^{-1}(\\frac{u}{a}) + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{du}{u\\sqrt{u^2 - a^2}} = \\frac{1}{|a|}\\sec^{-1}(\\frac{|u|}{a}) + C[\/latex]<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Key Concepts<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain Restrictions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Inverse trig functions have restricted domains<\/li>\n<li class=\"whitespace-normal break-words\">Solutions must respect these domain restrictions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Recognizing Integrands:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify integrands that match standard forms<\/li>\n<li class=\"whitespace-normal break-words\">Use substitution to transform integrals into these forms<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship to Derivatives:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">These integrals are related to derivatives of inverse trig functions<\/li>\n<li class=\"whitespace-normal break-words\">Understanding this connection aids in application<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Handling Negative Integrands:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor out [latex]-1[\/latex] from negative integrands<\/li>\n<li class=\"whitespace-normal break-words\">Use existing formulas rather than memorizing new ones<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitution Techniques:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Often necessary to transform integrals into standard forms<\/li>\n<li class=\"whitespace-normal break-words\">May involve adjusting constants or variables<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1-16{x}^{2}}}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4782055\">Hint<\/button><\/p>\n<div id=\"q4782055\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572089952\">Substitute [latex]u=4x[\/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-id1170572480280\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572480280\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{4}\\phantom{\\rule{0.05em}{0ex}}{ \\sin }^{-1}(4x)+C[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/yVwFRhfDP_Y?controls=0&amp;start=173&amp;end=278&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.7IntegralsResultingInInverseTrigonometricFunctions173to278_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.7 Integrals Resulting in Inverse Trigonometric Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the indefinite integral using an inverse trigonometric function and substitution for [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{9-{x}^{2}}}.[\/latex]<\/p>\n<p id=\"fs-id1170572557808\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572557808\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572557808\" class=\"hidden-answer\" style=\"display: none\">[latex]{ \\sin }^{-1}\\left(\\frac{x}{3}\\right)+C[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/yVwFRhfDP_Y?controls=0&amp;start=365&amp;end=404&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.7IntegralsResultingInInverseTrigonometricFunctions365to404_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.7 Integrals Resulting in Inverse Trigonometric Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use substitution to find the antiderivative of [latex]\\displaystyle\\int \\frac{dx}{25+4{x}^{2}}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571562665\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571562665\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571562665\">[latex]\\frac{1}{10}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}\\left(\\frac{2x}{5}\\right)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]\\displaystyle\\int \\frac{dx}{16+{x}^{2}}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571609481\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571609481\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571609481\">[latex]\\frac{1}{4}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{x}{4})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572369374\">Evaluate the definite integral [latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{dx}{4+{x}^{2}}.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572176762\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572176762\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572176762\">[latex]\\frac{\\pi }{8}[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/yVwFRhfDP_Y?controls=0&amp;start=973&amp;end=1054&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.7IntegralsResultingInInverseTrigonometricFunctions973to1054_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.7 Integrals Resulting in Inverse Trigonometric Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":16,"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\/415"}],"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\/415\/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\/415\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=415"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=415"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=415"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}