{"id":1972,"date":"2021-08-19T16:03:43","date_gmt":"2021-08-19T16:03:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/review-for-integration-formulas-and-the-net-change-theorem\/"},"modified":"2025-11-05T19:20:46","modified_gmt":"2025-11-05T19:20:46","slug":"review-for-integration-formulas-and-the-net-change-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/review-for-integration-formulas-and-the-net-change-theorem\/","title":{"raw":"Skills Review for Integration Formulas and the Net Change Theorem","rendered":"Skills Review for Integration Formulas and the Net Change Theorem"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Divide two fractions<\/li>\r\n \t<li>Divide a fraction by a whole number<\/li>\r\n \t<li>Determine whether a function is even, odd, or neither<\/li>\r\n<\/ul>\r\n<\/div>\r\nttestIn the Integration Formulas and the Net Change Theorem section, we will need skills that include how to simplify complex fractions and determine the symmetry of a function. These skills are reviewed here.\r\n<h2>Simplify Complex Fractions<\/h2>\r\nWhen integrating, it is not uncommon to have to work with a complex fraction, that is, a fraction that contains fractions in its numerator\/denominator. An easy way to remember how to divide fractions is to multiply by the reciprocal. This means that you keep\u00a0the first number as-is, change the division sign to multiplication, and then find the reciprocal of the second number (flip it!).\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Complex Fractions<\/h3>\r\nSimplify [latex]\\dfrac{\\frac{2}{3}}{\\frac{1}{6}}[\/latex].\r\n\r\n[reveal-answer q=\"569112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"569112\"]\r\n\r\nNote that we can rewrite the given complex fraction as:\u00a0[latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}[\/latex]\r\n\r\nNow, multiply by the receprocal.\r\n\r\nKeep [latex]\\dfrac{2}{3}[\/latex]\r\n\r\nChange\u00a0 [latex] \\div [\/latex] to \u00a0[latex]\\cdot[\/latex]\r\n\r\nFlip [latex]\\dfrac{1}{6}[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\dfrac{2}{3}\\cdot\\dfrac{6}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{2\\cdot6}{3\\cdot1}=\\dfrac{12}{3}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{12}{3}=\\normalsize 4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Complex Fractions<\/h3>\r\nDivide [latex]\\dfrac{\\frac{3}{5}}{2}[\/latex].\r\n\r\n[reveal-answer q=\"950676\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"950676\"]\r\n\r\nNote that we can rewrite the given complex fraction as:\u00a0[latex]\\dfrac{3}{5}\\div\\dfrac{2}{1}[\/latex]\r\n\r\nMultiply by the reciprocal.\r\n\r\nKeep [latex]\\dfrac{3}{5}[\/latex], change [latex] \\div [\/latex] to [latex]\\cdot[\/latex], and flip [latex]2[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\dfrac{3}{5}\\cdot\\dfrac{1}{2}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{3\\cdot 1}{5\\cdot 2}=\\dfrac{3}{10}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]\\dfrac{\\frac{5}{7}}{\\frac{3}{8}}[\/latex].\r\n\r\n[reveal-answer q=\"133760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133760\"]\r\n\r\n[latex]\\dfrac{40}{21}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\r\nSome functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\\left(x\\right)={x}^{2}[\/latex] or [latex]f\\left(x\\right)=|x|[\/latex] will result in the original graph. We say that these types of graphs are symmetric about the [latex]y[\/latex]-axis. Functions whose graphs are symmetric about the <em>y<\/em>-axis are called <strong>even functions.<\/strong>\r\n\r\nIf the graphs of [latex]f\\left(x\\right)={x}^{3}[\/latex] or [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex] were reflected over <em>both<\/em> axes, the result would be the original graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203605\/CNX_Precalc_Figure_01_05_021abc2.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/> (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.[\/caption]\r\n\r\nWe say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an <strong>odd function<\/strong>.\r\n\r\nNote: A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\\left(x\\right)={2}^{x}[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\\left(x\\right)=0[\/latex].\r\n\r\n<iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6454976&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VvUI6E78cN4&amp;video_target=tpm-plugin-saqsqzpb-VvUI6E78cN4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/IntroductionToOddAndEvenFunctions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \"Introduction to Odd and Even Functions\" here (opens in new window)<\/a>.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Even and Odd Functions<\/h3>\r\nA function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex]\r\n\r\n[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]\r\n\r\nThe graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.\r\n\r\nA function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex]\r\n\r\n[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]\r\n\r\nThe graph of an odd function is symmetric about the origin.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the formula for a function, determine if the function is even, odd, or neither.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Determine whether the function satisfies [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]. If it does, it is even.<\/li>\r\n \t<li>Determine whether the function satisfies [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]. If it does, it is odd.<\/li>\r\n \t<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining whether a Function Is Even, Odd, or Neither<\/h3>\r\nIs the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?\r\n\r\n[reveal-answer q=\"936347\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"936347\"]\r\n\r\nWithout looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.\r\n<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\r\nThis does not return us to the original function, so this function is not even. We can now test the rule for odd functions.\r\n<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\r\nBecause [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.\r\n<h4>Analysis of the Solution<\/h4>\r\nConsider the graph of [latex]f[\/latex]. Notice that the graph is symmetric about the origin. For every point [latex]\\left(x,y\\right)[\/latex] on the graph, the corresponding point [latex]\\left(-x,-y\\right)[\/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f[\/latex], and the corresponding point [latex]\\left(-1,-3\\right)[\/latex] is also on the graph.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203607\/CNX_Precalc_Figure_01_05_0392.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"731\" height=\"488\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIs the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?\r\n\r\n[reveal-answer q=\"630369\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"630369\"]\r\n\r\nEven\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]112703[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Divide two fractions<\/li>\n<li>Divide a fraction by a whole number<\/li>\n<li>Determine whether a function is even, odd, or neither<\/li>\n<\/ul>\n<\/div>\n<p>ttestIn the Integration Formulas and the Net Change Theorem section, we will need skills that include how to simplify complex fractions and determine the symmetry of a function. These skills are reviewed here.<\/p>\n<h2>Simplify Complex Fractions<\/h2>\n<p>When integrating, it is not uncommon to have to work with a complex fraction, that is, a fraction that contains fractions in its numerator\/denominator. An easy way to remember how to divide fractions is to multiply by the reciprocal. This means that you keep\u00a0the first number as-is, change the division sign to multiplication, and then find the reciprocal of the second number (flip it!).<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Complex Fractions<\/h3>\n<p>Simplify [latex]\\dfrac{\\frac{2}{3}}{\\frac{1}{6}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q569112\">Show Solution<\/span><\/p>\n<div id=\"q569112\" class=\"hidden-answer\" style=\"display: none\">\n<p>Note that we can rewrite the given complex fraction as:\u00a0[latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}[\/latex]<\/p>\n<p>Now, multiply by the receprocal.<\/p>\n<p>Keep [latex]\\dfrac{2}{3}[\/latex]<\/p>\n<p>Change\u00a0 [latex]\\div[\/latex] to \u00a0[latex]\\cdot[\/latex]<\/p>\n<p>Flip [latex]\\dfrac{1}{6}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{2}{3}\\cdot\\dfrac{6}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{2\\cdot6}{3\\cdot1}=\\dfrac{12}{3}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{12}{3}=\\normalsize 4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Complex Fractions<\/h3>\n<p>Divide [latex]\\dfrac{\\frac{3}{5}}{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q950676\">Show Solution<\/span><\/p>\n<div id=\"q950676\" class=\"hidden-answer\" style=\"display: none\">\n<p>Note that we can rewrite the given complex fraction as:\u00a0[latex]\\dfrac{3}{5}\\div\\dfrac{2}{1}[\/latex]<\/p>\n<p>Multiply by the reciprocal.<\/p>\n<p>Keep [latex]\\dfrac{3}{5}[\/latex], change [latex]\\div[\/latex] to [latex]\\cdot[\/latex], and flip [latex]2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{3}{5}\\cdot\\dfrac{1}{2}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{3\\cdot 1}{5\\cdot 2}=\\dfrac{3}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]\\dfrac{\\frac{5}{7}}{\\frac{3}{8}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133760\">Show Solution<\/span><\/p>\n<div id=\"q133760\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{40}{21}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\n<p>Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\\left(x\\right)={x}^{2}[\/latex] or [latex]f\\left(x\\right)=|x|[\/latex] will result in the original graph. We say that these types of graphs are symmetric about the [latex]y[\/latex]-axis. Functions whose graphs are symmetric about the <em>y<\/em>-axis are called <strong>even functions.<\/strong><\/p>\n<p>If the graphs of [latex]f\\left(x\\right)={x}^{3}[\/latex] or [latex]f\\left(x\\right)=\\dfrac{1}{x}[\/latex] were reflected over <em>both<\/em> axes, the result would be the original graph.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203605\/CNX_Precalc_Figure_01_05_021abc2.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/><\/p>\n<p class=\"wp-caption-text\">(a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.<\/p>\n<\/div>\n<p>We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an <strong>odd function<\/strong>.<\/p>\n<p>Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\\left(x\\right)={2}^{x}[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6454976&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VvUI6E78cN4&amp;video_target=tpm-plugin-saqsqzpb-VvUI6E78cN4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/IntroductionToOddAndEvenFunctions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;Introduction to Odd and Even Functions&#8221; here (opens in new window)<\/a>.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Even and Odd Functions<\/h3>\n<p>A function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex]<\/p>\n<p>[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<p>The graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.<\/p>\n<p>A function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex]<\/p>\n<p>[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]<\/p>\n<p>The graph of an odd function is symmetric about the origin.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the formula for a function, determine if the function is even, odd, or neither.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Determine whether the function satisfies [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]. If it does, it is even.<\/li>\n<li>Determine whether the function satisfies [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]. If it does, it is odd.<\/li>\n<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining whether a Function Is Even, Odd, or Neither<\/h3>\n<p>Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q936347\">Show Solution<\/span><\/p>\n<div id=\"q936347\" class=\"hidden-answer\" style=\"display: none\">\n<p>Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\n<p>This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.<\/p>\n<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\n<p>Because [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Consider the graph of [latex]f[\/latex]. Notice that the graph is symmetric about the origin. For every point [latex]\\left(x,y\\right)[\/latex] on the graph, the corresponding point [latex]\\left(-x,-y\\right)[\/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f[\/latex], and the corresponding point [latex]\\left(-1,-3\\right)[\/latex] is also on the graph.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203607\/CNX_Precalc_Figure_01_05_0392.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"731\" height=\"488\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q630369\">Show Solution<\/span><\/p>\n<div id=\"q630369\" class=\"hidden-answer\" style=\"display: none\">\n<p>Even<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm112703\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=112703&theme=oea&iframe_resize_id=ohm112703&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1972\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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