{"id":23,"date":"2024-04-16T20:38:56","date_gmt":"2024-04-17T00:38:56","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/?post_type=chapter&#038;p=23"},"modified":"2025-07-08T14:27:38","modified_gmt":"2025-07-08T18:27:38","slug":"indefinite-integrals-antiderivatives","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/chapter\/indefinite-integrals-antiderivatives\/","title":{"raw":"Unit 2: Indefinite Integrals: Antiderivatives","rendered":"Unit 2: Indefinite Integrals: Antiderivatives"},"content":{"raw":"In integral calculus, indefinite integrals play a fundamental role in finding antiderivatives and understanding the accumulated change of a function over an interval. Let's delve into the concept of indefinite integrals, their connection to antiderivatives, and how to utilize a table of indefinite integrals effectively for problem-solving.\r\n<ul>\r\n \t<li>What are Indefinite Integrals?<\/li>\r\n<\/ul>\r\nIndefinite integrals, denoted by [latex]\\int f(x) dx[\/latex], represent <strong>a family of functions<\/strong> rather than a single numerical value. They are expressed using the integral symbol without specified upper and lower limits. <span> From a mathematical perspective,<\/span> an indefinite integral represents all possible antiderivatives of a given function [latex]f(x)[\/latex]. In simpler terms, it answers the question: \"What function has a derivative equal to [latex]f(x)[\/latex]?\"\u00a0 <span>For instance, a set of functions<\/span>, [latex]x^2-100, ..., x^2, ..., x^2+100 [\/latex]) <span>all share the derivative<\/span> [latex]2x[\/latex], hence [latex]\\int 2x dx=x^2+C[\/latex] where C is any real number.\r\n<ul>\r\n \t<li>Relationship with Antiderivatives<\/li>\r\n<\/ul>\r\nIndefinite integrals are closely related to antiderivatives. An antiderivative of <strong>a function<\/strong> [latex]f(x)[\/latex] is any function [latex]F(x)[\/latex] whose derivative equals [latex]f(x)[\/latex]. Therefore, an indefinite integral [latex]\\int f(x) dx[\/latex] represents the general antiderivative of [latex]f(x)[\/latex]. Symbolically, if [latex]F(x)[\/latex] is an antiderivative of f(x), then[latex]\\int f(x) dx = F(x) + C[\/latex], where [latex]C[\/latex] is the constant of integration.\r\n<ul>\r\n \t<li>Table of Indefinite Integrals:<\/li>\r\n<\/ul>\r\nA table of indefinite integrals provides a comprehensive list of common functions and their corresponding antiderivatives. Some essential entries in this table include:<span><\/span>\r\n<p style=\"padding-left: 120px;\"><img src=\"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/04\/Table_derivatives_integrals-1-300x244.png\" alt=\"This table is a comparison table of derivatives and integrals\" width=\"423\" height=\"344\" class=\"alignnone wp-image-68\" \/><\/p>\r\n\r\n<div class=\"flex-1 overflow-hidden\">\r\n<div class=\"react-scroll-to-bottom--css-oovfj-79elbk h-full\">\r\n<div class=\"react-scroll-to-bottom--css-oovfj-1n7m0yu\">\r\n<div class=\"flex flex-col text-sm pb-9\">\r\n<div class=\"w-full text-token-text-primary\" dir=\"auto\" data-testid=\"conversation-turn-141\">\r\n<div class=\"px-4 py-2 justify-center text-base md:gap-6 m-auto\">\r\n<div class=\"flex flex-1 text-base mx-auto gap-3 juice:gap-4 juice:md:gap-6 md:px-5 lg:px-1 xl:px-5 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\">\r\n<div class=\"relative flex w-full flex-col agent-turn\">\r\n<div class=\"flex-col gap-1 md:gap-3\">\r\n<div class=\"flex flex-grow flex-col max-w-full\">\r\n<div data-message-author-role=\"assistant\" data-message-id=\"99869da1-15ed-449c-a823-12a52fc37d07\" dir=\"auto\" class=\"min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&amp;]:mt-5 overflow-x-auto\">\r\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\r\n<h2><strong>Practice Questions:<\/strong><\/h2>\r\nEvaluate the following indefinite integrals before referring to the video for comprehensive solutions.\r\n\r\nQ1. [latex]\\int e \\,dx[\/latex]\r\n\r\nQ2. [latex]\\int -4.1 \\,dx[\/latex]\r\n\r\nQ3. [latex]\\int 0 \\,dx[\/latex]\r\n\r\nQ4. [latex]\\int x^2 \\,dx[\/latex]\r\n\r\n[video width=\"1854\" height=\"1176\" mp4=\"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/11\/Unit2_partA.mp4\"][\/video]\r\n\r\nQ5. [latex]\\int x^{2\/3} \\, dx[\/latex]\r\n\r\nQ6. [latex]\\int x^{-\\pi} \\,dx[\/latex]\r\n\r\nQ7. [latex]\\int \\sqrt{x^3} \\,dx[\/latex]\r\n\r\nQ8. [latex]\\int \\sqrt[5]{x^2} \\,dx[\/latex]\r\n\r\nQ9. [latex]\\int \\frac{1}{x^2} \\,dx[\/latex]\r\n\r\nQ10. [latex]\\int \\frac{1}{\\sqrt[3]{x^2}} \\,dx[\/latex]\r\n\r\nQ11. [latex]\\int 5^t \\,dt[\/latex]\r\n\r\nQ12. [latex]\\int \\sin\\theta \\,d\\theta[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<p>In integral calculus, indefinite integrals play a fundamental role in finding antiderivatives and understanding the accumulated change of a function over an interval. Let&#8217;s delve into the concept of indefinite integrals, their connection to antiderivatives, and how to utilize a table of indefinite integrals effectively for problem-solving.<\/p>\n<ul>\n<li>What are Indefinite Integrals?<\/li>\n<\/ul>\n<p>Indefinite integrals, denoted by [latex]\\int f(x) dx[\/latex], represent <strong>a family of functions<\/strong> rather than a single numerical value. They are expressed using the integral symbol without specified upper and lower limits. <span> From a mathematical perspective,<\/span> an indefinite integral represents all possible antiderivatives of a given function [latex]f(x)[\/latex]. In simpler terms, it answers the question: &#8220;What function has a derivative equal to [latex]f(x)[\/latex]?&#8221;\u00a0 <span>For instance, a set of functions<\/span>, [latex]x^2-100, ..., x^2, ..., x^2+100[\/latex]) <span>all share the derivative<\/span> [latex]2x[\/latex], hence [latex]\\int 2x dx=x^2+C[\/latex] where C is any real number.<\/p>\n<ul>\n<li>Relationship with Antiderivatives<\/li>\n<\/ul>\n<p>Indefinite integrals are closely related to antiderivatives. An antiderivative of <strong>a function<\/strong> [latex]f(x)[\/latex] is any function [latex]F(x)[\/latex] whose derivative equals [latex]f(x)[\/latex]. Therefore, an indefinite integral [latex]\\int f(x) dx[\/latex] represents the general antiderivative of [latex]f(x)[\/latex]. Symbolically, if [latex]F(x)[\/latex] is an antiderivative of f(x), then[latex]\\int f(x) dx = F(x) + C[\/latex], where [latex]C[\/latex] is the constant of integration.<\/p>\n<ul>\n<li>Table of Indefinite Integrals:<\/li>\n<\/ul>\n<p>A table of indefinite integrals provides a comprehensive list of common functions and their corresponding antiderivatives. Some essential entries in this table include:<span><\/span><\/p>\n<p style=\"padding-left: 120px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/04\/Table_derivatives_integrals-1-300x244.png\" alt=\"This table is a comparison table of derivatives and integrals\" width=\"423\" height=\"344\" class=\"alignnone wp-image-68\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/04\/Table_derivatives_integrals-1-300x244.png 300w, https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/04\/Table_derivatives_integrals-1-65x53.png 65w, https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/04\/Table_derivatives_integrals-1-225x183.png 225w, https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/04\/Table_derivatives_integrals-1-350x285.png 350w, https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/04\/Table_derivatives_integrals-1.png 696w\" sizes=\"auto, (max-width: 423px) 100vw, 423px\" \/><\/p>\n<div class=\"flex-1 overflow-hidden\">\n<div class=\"react-scroll-to-bottom--css-oovfj-79elbk h-full\">\n<div class=\"react-scroll-to-bottom--css-oovfj-1n7m0yu\">\n<div class=\"flex flex-col text-sm pb-9\">\n<div class=\"w-full text-token-text-primary\" dir=\"auto\" data-testid=\"conversation-turn-141\">\n<div class=\"px-4 py-2 justify-center text-base md:gap-6 m-auto\">\n<div class=\"flex flex-1 text-base mx-auto gap-3 juice:gap-4 juice:md:gap-6 md:px-5 lg:px-1 xl:px-5 md:max-w-3xl lg:max-w-[40rem] xl:max-w-[48rem]\">\n<div class=\"relative flex w-full flex-col agent-turn\">\n<div class=\"flex-col gap-1 md:gap-3\">\n<div class=\"flex flex-grow flex-col max-w-full\">\n<div data-message-author-role=\"assistant\" data-message-id=\"99869da1-15ed-449c-a823-12a52fc37d07\" dir=\"auto\" class=\"min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&amp;]:mt-5 overflow-x-auto\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<h2><strong>Practice Questions:<\/strong><\/h2>\n<p>Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.<\/p>\n<p>Q1. [latex]\\int e \\,dx[\/latex]<\/p>\n<p>Q2. [latex]\\int -4.1 \\,dx[\/latex]<\/p>\n<p>Q3. [latex]\\int 0 \\,dx[\/latex]<\/p>\n<p>Q4. [latex]\\int x^2 \\,dx[\/latex]<\/p>\n<div style=\"width: 1854px;\" class=\"wp-video\"><!--[if lt IE 9]><script>document.createElement('video');<\/script><![endif]--><br \/>\n<video class=\"wp-video-shortcode\" id=\"video-23-1\" width=\"1854\" height=\"1176\" preload=\"metadata\" controls=\"controls\"><source type=\"video\/mp4\" src=\"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/11\/Unit2_partA.mp4?_=1\" \/><a href=\"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/11\/Unit2_partA.mp4\">https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-content\/uploads\/sites\/420\/2024\/11\/Unit2_partA.mp4<\/a><\/video><\/div>\n<p>Q5. [latex]\\int x^{2\/3} \\, dx[\/latex]<\/p>\n<p>Q6. [latex]\\int x^{-\\pi} \\,dx[\/latex]<\/p>\n<p>Q7. [latex]\\int \\sqrt{x^3} \\,dx[\/latex]<\/p>\n<p>Q8. [latex]\\int \\sqrt[5]{x^2} \\,dx[\/latex]<\/p>\n<p>Q9. [latex]\\int \\frac{1}{x^2} \\,dx[\/latex]<\/p>\n<p>Q10. [latex]\\int \\frac{1}{\\sqrt[3]{x^2}} \\,dx[\/latex]<\/p>\n<p>Q11. [latex]\\int 5^t \\,dt[\/latex]<\/p>\n<p>Q12. [latex]\\int \\sin\\theta \\,d\\theta[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":391,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-23","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/23","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/users\/391"}],"version-history":[{"count":22,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/23\/revisions"}],"predecessor-version":[{"id":283,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/23\/revisions\/283"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/23\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/media?parent=23"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapter-type?post=23"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/contributor?post=23"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/license?post=23"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}