{"id":32647,"date":"2023-04-04T13:00:22","date_gmt":"2023-04-04T13:00:22","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32647"},"modified":"2025-03-27T11:56:11","modified_gmt":"2025-03-27T11:56:11","slug":"indefinite-integrals-and-basic-integration-techniques","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/indefinite-integrals-and-basic-integration-techniques\/","title":{"rendered":"Indefinite Integrals and Basic Integration Techniques"},"content":{"rendered":"<style>\n    p, ul, ol {\n        text-align: justify;\n    }\n    h1, h2, h3 {\n    text-align:center;\n    }\n<\/style>\n<p><center><\/p>\n<h1>Indefinite Integrals and Basic Integration Techniques<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\">This class introduces the basic techniques to compute the most elementary indefinite integrals, as well as the properties of the integration operator. This includes polynomial, exponential, hyperbolic, and basic trigonometric integrals.<\/p>\n<p style=\"text-align:center;\"><strong><u>Learning Objectives<\/u>:<\/strong><br \/>By the end of this class, the student will be able to<\/p>\n<ol>\n<li><strong>Understand<\/strong> the process of indefinite integration as the inverse process of differentiation.<\/li>\n<li><strong>Compute<\/strong> the integral of polynomials and expressions involving exponential, hyperbolic, and trigonometric functions.<\/li>\n<li><strong>Use<\/strong> the properties of integrals to perform algebraic manipulations that facilitate their computation.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>TABLE OF CONTENTS<\/strong><br \/>\n<a href=\"#1\">THE RELEVANCE OF INDEFINITE INTEGRALS<\/a><br \/>\n<a href=\"#2\">ANTIDERIVATIVES, INDEFINITE INTEGRALS, AND FUNCTION PRIMITIVES<\/a><br \/>\n<a href=\"#3\">BASIC INTEGRATION TECHNIQUES<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/4wSTxA7zY9k\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>The Relevance of Indefinite Integrals<\/h2>\n<p>Indefinite integrals are a fundamental tool in calculus and have a wide range of applications in physical and mathematical sciences. They allow us to compute the primitive function of a given function, which in turn is used to calculate areas under curves, volumes of solids, probabilities, and many other applications in physics, engineering, statistics, and economics. Furthermore, indefinite integrals are essential for solving differential equations, making them indispensable in many fields of science and technology.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/56fMLiVPwDI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"2\"><\/a><\/p>\n<h2>Antiderivatives, Indefinite Integrals, and Function Primitives<\/h2>\n<p>If a function <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> has derivative <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> on a given interval <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>, then <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> is said to be a primitive of <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> on that interval.<\/p>\n<p>It is important to note that if <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> is a primitive of <span class=\"katex-eq\" data-katex-display=\"false\">f(x),<\/span> then so is <span class=\"katex-eq\" data-katex-display=\"false\">F(x) + C,<\/span> where <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> is any real constant. This is represented by writing:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int f(x) dx = F(x) + C<\/span>\n<p>The constant <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> is known as the <strong>constant of integration<\/strong>, and its presence indicates that the primitive of a function is not a single function but a family of functions: the set of all functions whose derivative is <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> on the interval <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>.<\/p>\n<p>The words antiderivative, primitive, and indefinite integral are three ways to express the same idea, so we use them interchangeably. In summary, the indefinite integral is the inverse process of differentiation, and from this idea its most fundamental properties are derived.<\/p>\n<h3>Basic Properties of Indefinite Integrals<\/h3>\n<p>In order to compute indefinite integrals, we must first know some basic properties, which are directly inherited from the properties of derivatives.<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int  \\dfrac{df(x)}{dx} dx = f(x) + C<\/span><\/br>Because the indefinite integral is the inverse process of differentiation.<\/li>\n<p><\/br><\/p>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\lambda f(x) dx = \\lambda \\int f(x) dx<\/span><\/br>Where <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda<\/span> is any real constant. This holds because<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int \\lambda \\dfrac{d\\phi(x)}{dx}dx &amp;=  \\displaystyle \\int \\dfrac{d}{dx}\\lambda \\phi(x) dx \\\\ \\\\\n\n&amp;= \\lambda \\phi(x) + C_1 \\\\ \\\\\n\n&amp;= \\lambda(\\phi(x) + C_2) \\\\ \\\\\n\n&amp;= \\lambda \\displaystyle  \\int \\frac{d\\phi(x)}{dx}dx \\end{array}<\/span><\/center><br \/>\n<\/br><br \/>\nAnd then, using <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{d\\phi(x)}{dx}<\/span> we get<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\lambda f(x) dx = \\lambda \\int f(x)dx<\/span><\/center><\/li>\n<p><\/br><\/p>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int f(x) + g(x) dx = \\int f(x) dx + \\int g(x) dx <\/span>\n<\/br><br \/>\nThis can be demonstrated in a similar way. Consider two functions <span class=\"katex-eq\" data-katex-display=\"false\">\\phi(x)<\/span> and  <span class=\"katex-eq\" data-katex-display=\"false\">\\psi(x)<\/span> such that<br \/>\n<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{d\\phi(x)}{dx}<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\dfrac{d\\psi(x)}{dx}<\/span><\/center><br \/>\n<\/br><br \/>\nThen we have<br \/>\n<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int f(x) + g(x) dx\n\n&amp;= \\displaystyle \\int \\dfrac{d\\phi(x)}{dx} +  \\dfrac{d\\psi(x)}{dx} dx \\\\ \\\\\n\n&amp;= \\displaystyle \\int \\dfrac{d}{dx} (\\phi(x)  + \\psi(x)) dx \\\\ \\\\\n\n&amp;= \\phi(x) + \\psi(x) + C \\\\ \\\\\n\n&amp;= (\\phi(x) + C_1) + (\\psi(x) + C_2) \\\\ \\\\\n\n&amp;= \\displaystyle \\int \\dfrac{d\\phi(x)}{dx} dx + \\int \\dfrac{d\\psi(x)}{dx}dx \\\\ \\\\\n\n&amp;= \\displaystyle \\int f(x) dx + \\int g(x) dx\n\n\\end{array}<\/span><\/center>\n<\/li>\n<\/ol>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>Basic Integration Techniques<\/h2>\n<p>There are basic integration techniques that allow us to compute some indefinite integrals using results obtained from differentiation. Through these techniques, we can obtain the following useful results for integration:<\/p>\n<h3>Integrals of Polynomial Functions<\/h3>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int 1 dx = x + C<\/span>\n<\/br><br \/>\nBecause  <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} (x + C)= 1 <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int x^q dx = \\dfrac{x^{q+1}}{q+1}  + C,<\/span> provided that <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq -1<\/span>\n<\/br><br \/>\nBecause <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left(\\dfrac{x^{q+1}}{q+1}  + C\\right) = x^q.<\/span>\n<\/li>\n<\/ol>\n<p>With these results and the basic properties, we can compute the integral of any polynomial without difficulty.<\/p>\n<div style=\"background-color:#F3FFF3; padding:20px;\">\n<p><strong>Example:<\/strong><\/p>\n<ol>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 3x+2 \\right) dx =  \\dfrac{3}{2}x^2 + 2x + C<\/span><\/li>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 5x^2 + 2x + 3 \\right) dx= \\dfrac{5}{3}x^3 + x + 3x  + C<\/span><\/li>\n<li type=\"a\"> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 4x^{12} - 7x^{-1\/3} + 1 \\right) dx  <\/span> <\/li>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} &amp;= \\dfrac{4}{13}x^{13} - \\dfrac{7}{2\/3}x^{2\/3} + x + C \\\\ \\\\\n\n&amp;= \\dfrac{4}{13}x^{13} - \\dfrac{21}{2}x^{2\/3} + x + C\n\n\\end{array}<\/span>\n<\/ol>\n<\/div>\n<h3>Exponential and Logarithmic Integrals<\/h3>\n<p>From the known results of the derivatives of exponential and logarithmic functions, we obtain the following basic results:<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int e^{x}dx = e^{x} + C<\/span>\n<br \/>\nBecause <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left(e^x + C\\right) = e^x<\/span>\n<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{1}{x} dx = ln|x| + C<\/span>\n<\/br><br \/>\nBecause <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left(ln|x| + C \\right) = \\dfrac{1}{|x|} sig(x) = \\dfrac{1}{x}<\/span>\n<\/br><br \/>\nWhere <span class=\"katex-eq\" data-katex-display=\"false\">sig(x)<\/span> is the sign function defined as follows:<br \/>\n<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">sig(x) = \\left\\{\\begin{array}{} +1 &amp;,&amp;0\\lt x \\\\ -1 &amp;,&amp; x\\lt 0 \\end{array}\\right.<\/span><\/center>\n<\/li>\n<\/ol>\n<p>The result of the integral of <span class=\"katex-eq\" data-katex-display=\"false\">1\/x<\/span> allows us to expand our ability to integrate functions, as we can begin integrating functions that are quotients of polynomials.<\/p>\n<div style=\"background-color:#F3FFF3; padding:20px;\">\n<p><strong>Example:<\/strong><\/p>\n<ol>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{x^2 + 3x + 2}{5x^2}dx = \\int \\dfrac{1}{5} + \\dfrac{3}{5}\\dfrac{1}{x} + \\dfrac{2}{5}\\dfrac{1}{x^2}dx<\/span>\n<\/br><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\dfrac{x}{5}+\\dfrac{3}{5}ln(x) - \\dfrac{2}{5}\\dfrac{1}{x} + C <\/span><\/li>\n<p><\/br><\/p>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{x^2 - 3 x + 2}{(x-2)^2}dx = \\int \\dfrac{(x-2)^2 + (x-2)}{(x-2)^2} dx<\/span><\/li>\n<p><\/br><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">= \\displaystyle \\int 1 + \\dfrac{1}{x-2} dx\\\\ \\\\\n\n= x + \\displaystyle \\int \\dfrac{1}{x-2}dx = x + ln|x-2| + C<\/span>\n<\/br><br \/>\nBecause<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left( ln|x-2| + C\\right) = \\dfrac{1}{|x-2|}sig(x-2) = \\dfrac{1}{x-2}<\/span>\n<\/ol>\n<\/div>\n<h3>Integrals of Basic Hyperbolic Functions<\/h3>\n<p>The basic hyperbolic functions are<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} sinh(x) &amp;=&amp; \\dfrac{e^x - e^{-x}}{2} \\\\ \\\\\n\ncosh(x) &amp;=&amp; \\dfrac{e^x + e^{-x}}{2}\n\n\\end{array}<\/span>\n<p>Since we have already seen how the exponential function integral works, we will have no problem with the integrals of hyperbolic sine and cosine.<\/p>\n<p>For hyperbolic sine the computation is practically direct:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int sinh(x) dx\n\n&amp;=&amp; \\displaystyle \\int \\dfrac{e^x - e^{-x}}{2}dx \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left( \\displaystyle \\int e^x dx - \\int e^{-x}  dx \\right) \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left(e^x + e^{-x} \\right) + C = cosh(x) + C\n\n\\end{array}<\/span>\n<p>And for hyperbolic cosine, the calculations are practically analogous:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int cosh(x) dx\n\n&amp;=&amp; \\displaystyle \\int \\dfrac{e^x + e^{-x}}{2}dx \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left( \\displaystyle \\int e^x dx + \\int e^{-x}  dx \\right) \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left(e^x - e^{-x} \\right) + C = sinh(x) + C\n\n\\end{array}<\/span>\n<p>Besides these, there are many other hyperbolic functions that can be integrated:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} tanh(x) &amp;=&amp; \\dfrac{sinh(x)}{cosh(x)} \\\\\n\nsech(x) &amp;=&amp; \\dfrac{1}{cosh(x)} \\\\\n\n{}csch(x) &amp;=&amp; \\dfrac{1}{sinh(x)} \\\\\n\nctgh(x) &amp;=&amp; \\dfrac{1}{tanh(x)}\n\n\\end{array}<\/span>\n<p>However, integrating them requires other techniques that we will explore in future classes.<\/p>\n<h3>Integrals of Basic Trigonometric Functions<\/h3>\n<p>The basic trigonometric functions are <span class=\"katex-eq\" data-katex-display=\"false\">sin(x)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">cos(x)<\/span>. Computing their integrals is practically direct based on what we already know from their derivatives.<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int sin(x) dx = -cos(x) + C \\\\ \\\\\n\n{} \\displaystyle \\int cos(x) dx = sen(x) + C\n\n\\end{array}<\/span>\n<p>This occurs because<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{}  \\dfrac{d}{dx}\\left( sin(x) + C \\right) &amp;=&amp; cos(x) \\\\ \\\\\n\n{}  \\dfrac{d}{dx}\\left( cos(x) + C \\right) &amp;=&amp; -sin(x) \\\\ \\\\\n\n\\end{array}<\/span>\n<h2>Conclusion<\/h2>\n<p>In this class, we have explored indefinite integrals from their theoretical foundations to their most basic practical applications. We have learned to recognize them as the inverse process of differentiation, to identify their basic properties, and to apply direct techniques to integrate simple polynomial, exponential, logarithmic, hyperbolic, and trigonometric functions. This knowledge forms the essential foundation for tackling more complex integration problems in the future and will be fundamental for studying advanced applications in physics, engineering, and other sciences. With this foundational knowledge, we will be able to introduce more sophisticated techniques in future classes.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Indefinite Integrals and Basic Integration Techniques This class introduces the basic techniques to compute the most elementary indefinite integrals, as well as the properties of the integration operator. This includes polynomial, exponential, hyperbolic, and basic trigonometric integrals. Learning Objectives:By the end of this class, the student will be able to Understand the process of indefinite [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32629,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":14,"footnotes":""},"categories":[1133,567],"tags":[],"class_list":["post-32647","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-integral-calculus","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Indefinite Integrals and Basic Integration Techniques - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Master the basic integration techniques with clear examples and step-by-step explanations. 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