{"id":825525,"date":"2024-05-07T19:15:39","date_gmt":"2024-05-07T13:45:39","guid":{"rendered":"https:\/\/leverageedu.com\/discover\/?p=825525"},"modified":"2024-05-07T19:15:39","modified_gmt":"2024-05-07T13:45:39","slug":"basic-concepts-integration-formulas","status":"publish","type":"post","link":"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-integration-formulas\/","title":{"rendered":"Integration Formulas: Examples and Solutions"},"content":{"rendered":"\n<p>Integration is a fundamental concept in calculus that allows us to find the area under a curve or the accumulation of a quantity over an interval. The Integration Formula is a set of mathematical expressions that help us evaluate integrals efficiently. Simply put, Integration is like finding the total amount of something by adding up tiny pieces of them over a range. Similar to calculating the entire area under a curve by summing countless tiny rectangles that fit beneath it. Read on to learn more about the different integration formulas and their applications.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Difference Between Differentiation and Integration<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-integration-formula-by-parts\">Integration Formula by Parts<\/h2>\n\n\n\n<p>The Integration by Parts Formula is used to evaluate integrals where the integrand is a product of two functions, one of which is easy to differentiate, and the other is easy to integrate.<\/p>\n\n\n\n<p>The <strong>Formula<\/strong> is:<\/p>\n\n\n\n<p>\u222b u dv = uv &#8211; \u222b v du<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>u and dv are the first function and its differential, respectively.<\/li>\n\n\n\n<li>v and du are the second function and its differential, respectively.<\/li>\n<\/ul>\n\n\n\n<p><strong>Example 1:<\/strong><\/p>\n\n\n\n<p>Evaluate the integral \u222b x e^x dx<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Let u = x and dv = e^x dx<\/p>\n\n\n\n<p>Then, du = dx and v = e^x<\/p>\n\n\n\n<p>Applying the Integration by Parts Formula:<\/p>\n\n\n\n<p>\u222b x e^x dx = x e^x &#8211; \u222b e^x dx<\/p>\n\n\n\n<p>\u222b x e^x dx = x e^x &#8211; e^x + C<\/p>\n\n\n\n<p><strong>Example 2:<\/strong><\/p>\n\n\n\n<p>Evaluate the integral \u222b ln(x) dx<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Let u = ln(x) and dv = dx<\/p>\n\n\n\n<p>Then, du = 1\/x dx and v = x<\/p>\n\n\n\n<p>Applying the Integration by Parts Formula:<\/p>\n\n\n\n<p>\u222b ln(x) dx = x ln(x) &#8211; \u222b x (1\/x) dx<\/p>\n\n\n\n<p>\u222b ln(x) dx = x ln(x) &#8211; x + C<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Branches of Mathematics: Arithmetic, Algebra, Geometry, Calculus, Trigonometry, Topology, Probability and Statistics<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-integration-formula-by-substitution-nbsp\">Integration Formula by Substitution&nbsp;<\/h2>\n\n\n\n<p>The Integration by Substitution Formula, also known as the change of variable formula, is used to evaluate integrals where the integrand can be expressed in terms of a new variable.<\/p>\n\n\n\n<p>The <strong>Formula<\/strong> is:<\/p>\n\n\n\n<p>\u222b f(g(x)) g'(x) dx = \u222b f(u) du<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>g(x) is the new variable, and u = g(x)<\/li>\n\n\n\n<li>g'(x) is the derivative of g(x)<\/li>\n<\/ul>\n\n\n\n<p><strong>Example 1:<\/strong><\/p>\n\n\n\n<p>Evaluate the integral \u222b (2x + 3)^4 dx<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Let u = 2x + 3, then du = 2 dx<\/p>\n\n\n\n<p>Substituting, we get:<\/p>\n\n\n\n<p>\u222b (2x + 3)^4 dx = 1\/2 \u222b u^4 du<\/p>\n\n\n\n<p>\u222b (2x + 3)^4 dx = 1\/2 (u^5\/5) + C<\/p>\n\n\n\n<p>\u222b (2x + 3)^4 dx = 1\/2 ((2x + 3)^5\/5) + C<\/p>\n\n\n\n<p><strong>Example 2:<\/strong><\/p>\n\n\n\n<p>Evaluate the integral \u222b (1 + x^2)^(-1\/2) dx<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Let u = 1 + x^2, then du = 2x dx<\/p>\n\n\n\n<p>Substituting, we get:<\/p>\n\n\n\n<p>\u222b (1 + x^2)^(-1\/2) dx = 1\/2 \u222b u^(-1\/2) du<\/p>\n\n\n\n<p>\u222b (1 + x^2)^(-1\/2) dx = 1\/2 (2u^(1\/2)) + C<\/p>\n\n\n\n<p>\u222b (1 + x^2)^(-1\/2) dx = \u221a(1 + x^2) + C<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Profit and Loss Formula Questions<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-integration-formula-by-partial-fractions-nbsp\">Integration Formula by Partial Fractions&nbsp;<\/h2>\n\n\n\n<p>The Integration by Partial Fractions formula evaluates integrals where the integrand is a rational function, that is, a fraction with a polynomial in the numerator and a polynomial in the denominator.<\/p>\n\n\n\n<p>The general Steps are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Factorize the denominator of the rational function.<\/li>\n\n\n\n<li>Express the rational function as a sum of partial fractions.<\/li>\n\n\n\n<li>Integrate each partial fraction separately.<\/li>\n<\/ol>\n\n\n\n<p><strong>Example 1:<\/strong><\/p>\n\n\n\n<p>Evaluate the integral \u222b (x^2 + 3x + 2) \/ (x^2 &#8211; 1) dx<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Step 1: Factorize the denominator.<\/p>\n\n\n\n<p>x^2 &#8211; 1 = (x &#8211; 1)(x + 1)<\/p>\n\n\n\n<p>Step 2: Express the rational function as a sum of partial fractions.<\/p>\n\n\n\n<p>(x^2 + 3x + 2) \/ (x^2 &#8211; 1) = A \/ (x &#8211; 1) + B \/ (x + 1)<\/p>\n\n\n\n<p>Step 3: Integrate each partial fraction separately.<\/p>\n\n\n\n<p>\u222b (x^2 + 3x + 2) \/ (x^2 &#8211; 1) dx = \u222b A \/ (x &#8211; 1) dx + \u222b B \/ (x + 1) dx<\/p>\n\n\n\n<p>\u222b (x^2 + 3x + 2) \/ (x^2 &#8211; 1) dx = A ln|x &#8211; 1| + B ln|x + 1| + C<\/p>\n\n\n\n<p><strong>Example 2:<\/strong><\/p>\n\n\n\n<p>Evaluate the integral \u222b 1 \/ (x^3 &#8211; x) dx<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Step 1: Factorize the denominator.<\/p>\n\n\n\n<p>x^3 &#8211; x = x(x^2 &#8211; 1) = x(x &#8211; 1)(x + 1)<\/p>\n\n\n\n<p>Step 2: Express the rational function as a sum of partial fractions.<\/p>\n\n\n\n<p>1 \/ (x^3 &#8211; x) = A \/ x + B \/ (x &#8211; 1) + C \/ (x + 1)<\/p>\n\n\n\n<p>Step 3: Integrate each partial fraction separately.<\/p>\n\n\n\n<p>\u222b 1 \/ (x^3 &#8211; x) dx = \u222b A \/ x dx + \u222b B \/ (x &#8211; 1) dx + \u222b C \/ (x + 1) dx<\/p>\n\n\n\n<p>\u222b 1 \/ (x^3 &#8211; x) dx = A ln|x| + B ln|x &#8211; 1| &#8211; C ln|x + 1| + D<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Algebra Questions<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-definite-integration-formula\">Definite Integration Formula<\/h2>\n\n\n\n<p>The Definite Integration Formula is used to evaluate the integral of a function over a specific interval [a, b].<\/p>\n\n\n\n<p>The <strong>Formula<\/strong> is:<\/p>\n\n\n\n<p>\u222b_a^b f(x) dx = F(b) &#8211; F(a)<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>F(x) is the antiderivative or indefinite integral of f(x)<\/li>\n<\/ul>\n\n\n\n<p><strong>Example:<\/strong><\/p>\n\n\n\n<p>Evaluate the integral \u222b_0^2 (x^2 + 3x + 1) dx<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Step 1: Find the antiderivative of the integrand.<\/p>\n\n\n\n<p>F(x) = \u222b (x^2 + 3x + 1) dx = x^3\/3 + 3x^2\/2 + x + C<\/p>\n\n\n\n<p>Step 2: Evaluate the Definite Integral using the formula.<\/p>\n\n\n\n<p>\u222b_0^2 (x^2 + 3x + 1) dx = F(2) &#8211; F(0)<\/p>\n\n\n\n<p>\u222b_0^2 (x^2 + 3x + 1) dx = (2^3\/3 + 3(2)^2\/2 + 2 + C) &#8211; (0^3\/3 + 3(0)^2\/2 + 0 + C)<\/p>\n\n\n\n<p>\u222b_0^2 (x^2 + 3x + 1) dx = 8\/3 + 6 + 2 &#8211; 0 = 16\/3<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>What is the Difference Between Definite and Indefinite Integrals?<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-indefinite-integration-formula\">Indefinite Integration Formula<\/h2>\n\n\n\n<p>The Indefinite Integration Formula is used to find the antiderivative or primitive function of a given function.<\/p>\n\n\n\n<p>The <strong>Formula<\/strong> is:<\/p>\n\n\n\n<p>\u222b f(x) dx = F(x) + C<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>F(x) is the antiderivative or primitive function of f(x)<\/li>\n\n\n\n<li>C is the constant of integration<\/li>\n<\/ul>\n\n\n\n<p><strong>Example:<\/strong><\/p>\n\n\n\n<p>Evaluate the integral \u222b (2x + 3) \/ (x^2 + 1) dx<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Step 1: Identify the integrand.<\/p>\n\n\n\n<p>f(x) = (2x + 3) \/ (x^2 + 1)<\/p>\n\n\n\n<p>Step 2: Find the antiderivative of the integrand.<\/p>\n\n\n\n<p>\u222b (2x + 3) \/ (x^2 + 1) dx = \u222b 2 \/ (x^2 + 1) dx + \u222b 3 \/ (x^2 + 1) dx<\/p>\n\n\n\n<p>Using the Integration by Substitution Formula:<\/p>\n\n\n\n<p>\u222b 2 \/ (x^2 + 1) dx = 2 tan^-1(x) + C<\/p>\n\n\n\n<p>\u222b 3 \/ (x^2 + 1) dx = 3 tan^-1(x) + C<\/p>\n\n\n\n<p>Combining the results:<\/p>\n\n\n\n<p>\u222b (2x + 3) \/ (x^2 + 1) dx = 2 tan^-1(x) + 3 tan^-1(x) + C<\/p>\n\n\n\n<p>\u222b (2x + 3) \/ (x^2 + 1) dx = 5 tan^-1(x) + C<\/p>\n\n\n\n<p class=\"has-text-align-center has-vivid-red-color has-text-color has-link-color has-medium-font-size wp-elements-b9eb360a4bc4bdaa3c4feb84a1bd0d67\"><strong>Related Blogs\u00a0<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-types-of-fractions\/\"><strong>7 Types of Fractions with Examples<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-what-are-co-prime-numbers\/\"><strong>What are Co Prime Numbers?<\/strong><\/a><\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-how-to-find-percentage-of-marks\/\"><strong>How to Find Percentage of Marks?<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-tables-1-to-20\/\"><strong>Multiplication Tables of 1 to 20<\/strong><\/a><\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-ordinal-numbers\/\"><strong>Ordinal Numbers from 1 to 100!<\/strong><\/a><strong>&nbsp;<\/strong><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/civics-and-polity-table-of-17\/\"><strong>Table of 17: Multiples up to 20 &amp; a Trick!<\/strong><\/a>&nbsp;&nbsp;<\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-table-of-12\/\"><strong>Table of 12: Multiples up to 20!<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concept-hcf-of-two-consecutive-odd-numbers\/\"><strong>What is the HCF of Two Consecutive Odd Numbers?<\/strong><\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>I hope this helps! Did you like learning about the Integration Formula? Keep reading our blogs to learn more about the <a href=\"https:\/\/leverageedu.com\/discover\/category\/school-education\/basic-concepts\/\"><strong>basic concepts of Maths<\/strong><\/a>!<\/p>\n","protected":false},"excerpt":{"rendered":"Integration is a fundamental concept in calculus that allows us to find the area under a curve or&hellip;\n","protected":false},"author":106,"featured_media":825530,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"editor_notices":[],"footnotes":""},"categories":[423,476,389],"tags":[],"class_list":{"0":"post-825525","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-basic-concepts","8":"category-maths","9":"category-school-education"},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.3 (Yoast SEO v27.3) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Integration Formulas: Examples and Solutions - 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