{"id":687040,"date":"2023-09-11T15:57:35","date_gmt":"2023-09-11T10:27:35","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=687040"},"modified":"2025-07-25T17:27:35","modified_gmt":"2025-07-25T11:57:35","slug":"integration","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/articles\/integration\/","title":{"rendered":"Integration"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Introduction_to_Integration\" title=\"Introduction to Integration\">Introduction to Integration<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Definition_of_integration\" title=\"Definition of integration\">Definition of integration<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_In_math\" title=\"Integration In math\">Integration In math<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integral_calculus\" title=\"Integral calculus\">Integral calculus<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_-_Reverse_process_of_differentiation\" title=\"Integration \u2013 Reverse process of differentiation\">Integration \u2013 Reverse process of differentiation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Indefinite_integral\" title=\"Indefinite integral\">Indefinite integral<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Definite_integral\" title=\"Definite integral\">Definite integral<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integral_formulae_for_algebraic_functions\" title=\"Integral formulae for algebraic functions\">Integral formulae for algebraic functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Rules_for_integration\" title=\"Rules for integration\">Rules for integration<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integral_formulae_for_Trigonometric_functions\" title=\"Integral formulae for Trigonometric functions\">Integral formulae for Trigonometric functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_of_hyperbolic_function\" title=\"Integration of hyperbolic function\">Integration of hyperbolic function<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_of_rational_functions\" title=\"Integration of rational functions\">Integration of rational functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_of_irrational_function\" title=\"Integration of irrational function\">Integration of irrational function<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Special_integrals\" title=\"Special integrals\">Special integrals<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_by_substitution_related_formula\" title=\"Integration by substitution related formula\">Integration by substitution related formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_formula_by_partial_fractions\" title=\"Integration formula by partial fractions\">Integration formula by partial fractions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_by_parts\" title=\"Integration by parts\">Integration by parts<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Some_Integral_formulas_using_integration_by_parts\" title=\"Some Integral formulas using integration by parts\">Some Integral formulas using integration by parts<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Integration_formula_for_inverse_trigonometric_functions\" title=\"Integration formula for inverse trigonometric functions\">Integration formula for inverse trigonometric functions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Solve_some_examples_using_the_integration_formulas_mentioned_earlier\" title=\"Solve some examples using the integration formulas mentioned earlier\">Solve some examples using the integration formulas mentioned earlier<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Frequently_asked_questions_about_Integration_formulas\" title=\"Frequently asked questions about Integration formulas\">Frequently asked questions about Integration formulas<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#What_is_the_integration_formula\" title=\"What is the integration formula?\">What is the integration formula?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#What_are_the_4_types_of_integration\" title=\"What are the 4 types of integration?\">What are the 4 types of integration?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#Why_is_integration_used\" title=\"Why is integration used?\">Why is integration used?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#What_is_the_meaning_of_integration\" title=\"What is the meaning of integration?\">What is the meaning of integration?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#What_is_the_use_of_integration\" title=\"What is the use of integration?\">What is the use of integration?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/integration\/#What_is_integration\" title=\"What is integration?\">What is integration?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Introduction_to_Integration\"><\/span>Introduction to Integration<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Integration is a fundamental concept in calculus that deals with finding the antiderivative of a function. It involves calculating the accumulated change of a quantity over an interval. Integration allows us to determine areas under curves, volumes of solids, and solve diverse mathematical problems. It plays a vital role in various fields, including physics, engineering, economics, and is essential for advanced mathematical analysis..<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Definition_of_integration\"><\/span>Definition of integration<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>In calculus, integration is defined as the process of determining a function&#8217;s antiderivative. Integration is the inverse process of differentiation, and it includes determining the entire area under a curve or the accumulated change. The outcome of integration is a family of functions, each of which differs by a constant known as the constant of integration. Integration is essential in calculus since it allows for the computation of areas and volumes as well as the solution of a wide range of mathematical problems in numerous domains.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_In_math\"><\/span>Integration In math<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Integration is a fundamental subject in calculus that deals with determining a function&#8217;s antiderivative. It entails computing the total area or cumulative change under a curve. Integration is the inverse of differentiation in that the derivative of a function tells its rate of change, whereas the integral gives the original function when the rate of change is known.<\/p>\n<p>Integration is important in many domains of mathematics, science, engineering, and economics. It is used to compute areas, volumes, work, probability, and to answer a variety of mathematical issues that include continuous values. Integration is a strong technique for analysing function behaviour and comprehending the cumulative impacts of change..<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integral_calculus\"><\/span>Integral calculus<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Integral calculus is a branch of calculus that deals with computing areas under curves and determining the antiderivative of functions. It focuses on integration, which is the opposite of difference. Understanding cumulative amounts, volumes, effort, and numerous mathematical applications in science, engineering, and economics is impossible without integral calculus..<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_-_Reverse_process_of_differentiation\"><\/span>Integration \u2013 Reverse process of differentiation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li>Because integration is the inverse process of differentiation, we may discover the original function from which it was formed if we have the derivative.<\/li>\n<li>Consider differentiation as determining the rate of change or slope of a function at any point. In contrast, integration operates in the other direction. It collects information about the rate of change and discovers the original function that caused it.<\/li>\n<li>When we differentiate a function, we lose some information about the function&#8217;s constant term. Integration replaces the missing information with an arbitrary constant known as the constant of integration.<\/li>\n<li>If we differentiate the function f(x) = x<sup>2<\/sup>, for example, we obtain its derivative f'(x) = 2x. When we integrate 2x, we get the original function f(x) = x<sup>2<\/sup>, but we also need to add a constant C because we lost information about the constant term when differentiating.<\/li>\n<\/ul>\n<p>In summary, differentiation determines a function&#8217;s rate of change, whereas integration goes backward to get the original function from its rate of change. Differentiation and integration are essential procedures in calculus that allow us to understand function behaviour and solve a variety of mathematical issues.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Indefinite_integral\"><\/span>Indefinite integral<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Indefinite integrals, a concept in integral calculus, entail determining antiderivatives of functions that do not have particular integration limits. They provide a family of functions that differ by an arbitrary constant called the constant of integration. Indefinite integrals, denoted by the symbol, allow us to discover general solutions to differential equations and are an important tool for computing accumulated quantities and understanding the behaviour of functions..<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Definite_integral\"><\/span>Definite integral<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>In integral calculus, definite integrals require computing the cumulative area between a function&#8217;s curve and the x-axis over a certain interval. In contrast to indefinite integrals, definite integrals have upper and lower integration limits represented by &#8220;a&#8221; and &#8220;b,&#8221; respectively. The outcome of a definite integral is a single numerical number that reflects the net area encompassed by the curve. Definite integrals are used in mathematics, physics, and engineering to obtain total amounts, average values, and address numerous real-world situations..<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integral_formulae_for_algebraic_functions\"><\/span>Integral formulae for algebraic functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687043 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integral-formulae-for-algebraic-functions-.png\" alt=\"Integral formulae for algebraic functions\" width=\"297\" height=\"531\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integral-formulae-for-algebraic-functions-.png 297w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integral-formulae-for-algebraic-functions--168x300.png 168w\" sizes=\"(max-width: 297px) 100vw, 297px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Rules_for_integration\"><\/span>Rules for integration<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Here are some common integration formulas for algebraic functions:<\/p>\n<p><strong>Power Rule:<\/strong><\/p>\n<p>\u222b x^n dx = (x^(n+1))\/(n+1) + C, where n \u2260 -1.<\/p>\n<p><strong>Constant Multiple Rule:<\/strong><\/p>\n<p>\u222b k * f(x) dx = k * \u222b f(x) dx, where k is a constant.<\/p>\n<p><strong>Sum\/Difference Rule:<\/strong><\/p>\n<p>\u222b [f(x) + g(x)] dx = \u222b f(x) dx + \u222b g(x) dx.<\/p>\n<p><strong>Integration by Parts:<\/strong><\/p>\n<p>\u222b u dv = u * v &#8211; \u222b v du, where u and v are differentiable functions.<\/p>\n<p>These formulas are just a few examples of the wide range of algebraic functions that can be integrated. Integrating more complex functions may require using multiple techniques and approaches.<\/p>\n<p>&nbsp;<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integral_formulae_for_Trigonometric_functions\"><\/span>Integral formulae for Trigonometric functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687044 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integral-formulae-for-Trigonometric-functions.png\" alt=\"Integral formulae for Trigonometric functions\" width=\"553\" height=\"589\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integral-formulae-for-Trigonometric-functions.png 553w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integral-formulae-for-Trigonometric-functions-282x300.png 282w\" sizes=\"(max-width: 553px) 100vw, 553px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_of_hyperbolic_function\"><\/span><span class=\"TextRun SCXW196906298 BCX0\" lang=\"EN-IN\" xml:lang=\"EN-IN\" data-contrast=\"none\"><span class=\"NormalTextRun SCXW196906298 BCX0\" data-ccp-parastyle=\"heading 2\">Integration of hyperbolic function<\/span><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687045 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-hyperbolic-function-.png\" alt=\"Integration of hyperbolic function\" width=\"381\" height=\"589\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-hyperbolic-function-.png 381w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-hyperbolic-function--194x300.png 194w\" sizes=\"(max-width: 381px) 100vw, 381px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_of_rational_functions\"><\/span>Integration of rational functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687047 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-rational-functions.png\" alt=\"Integration of rational functions \" width=\"340\" height=\"379\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-rational-functions.png 340w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-rational-functions-269x300.png 269w\" sizes=\"(max-width: 340px) 100vw, 340px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_of_irrational_function\"><\/span>Integration of irrational function<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687048 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-irrational-function-.png\" alt=\"Integration of irrational function\" width=\"612\" height=\"805\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-irrational-function-.png 612w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-of-irrational-function--228x300.png 228w\" sizes=\"(max-width: 612px) 100vw, 612px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Special_integrals\"><\/span>Special integrals<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687049 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Special-integrals-.png\" alt=\"Special integrals \" width=\"427\" height=\"196\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Special-integrals-.png 427w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Special-integrals--300x138.png 300w\" sizes=\"(max-width: 427px) 100vw, 427px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_by_substitution_related_formula\"><\/span>Integration by substitution related formula<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687051 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-by-substitution-related-formula-.png\" alt=\"Integration by substitution related formula \" width=\"577\" height=\"481\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-by-substitution-related-formula-.png 577w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-by-substitution-related-formula--300x250.png 300w\" sizes=\"(max-width: 577px) 100vw, 577px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_formula_by_partial_fractions\"><\/span>Integration formula by partial fractions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687053 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-formula-by-partial-fractions-.png\" alt=\"Integration formula by partial fractions\" width=\"499\" height=\"505\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-formula-by-partial-fractions-.png 499w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-formula-by-partial-fractions--296x300.png 296w\" sizes=\"(max-width: 499px) 100vw, 499px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_by_parts\"><\/span>Integration by parts<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If f(x)og(x)dx are any two functions, then <strong>\u222bf(x).g(x)dx = \u222bf'(x)g(x)dxdx<\/strong><\/p>\n<p>Proper choice of first and second function<\/p>\n<ol>\n<li>The first function is the function which comes first in the word ILATE<\/li>\n<li>If one of the two functions is not directly integrable, then take this function as the first function.<\/li>\n<li>If one of the function is not directly integrable, and there is no other function, then unity is taken as the second function.<\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"Some_Integral_formulas_using_integration_by_parts\"><\/span>Some Integral formulas using integration by parts<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"wp-image-687056 size-full alignnone\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Some-Integral-formulas-using-integration-by-parts-1.png\" alt=\"\" width=\"468\" height=\"351\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Some-Integral-formulas-using-integration-by-parts-1.png 468w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Some-Integral-formulas-using-integration-by-parts-1-300x225.png 300w\" sizes=\"(max-width: 468px) 100vw, 468px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Integration_formula_for_inverse_trigonometric_functions\"><\/span>Integration formula for inverse trigonometric functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-687057 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-formula-for-inverse-trigonometric-functions-.png\" alt=\"Integration formula for inverse trigonometric functions\" width=\"463\" height=\"402\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-formula-for-inverse-trigonometric-functions-.png 463w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/09\/Integration-formula-for-inverse-trigonometric-functions--300x260.png 300w\" sizes=\"(max-width: 463px) 100vw, 463px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Solve_some_examples_using_the_integration_formulas_mentioned_earlier\"><\/span>Solve some examples using the integration formulas mentioned earlier<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Example 1:<\/strong> \u222b (3x^2 + 2x + 1) dx<\/p>\n<p>Using the Power Rule:<\/p>\n<p>\u222b (3x^2) dx = (3 * x^(2+1))\/(2+1) + C<\/p>\n<p>= (3\/3) * x^3 + C<\/p>\n<p>= x^3 + C.<\/p>\n<p><strong>Using the Power Rule:<\/strong><\/p>\n<p>\u222b (2x) dx = (2 * x^(1+1))\/(1+1) + C<\/p>\n<p>= x^2 + C.<\/p>\n<p><strong>Using the Power Rule:<\/strong><\/p>\n<p>\u222b (1) dx = x + C.<\/p>\n<p><strong>Putting it all together:<\/strong><\/p>\n<p>\u222b (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C.<\/p>\n<p><strong>Example 2:<\/strong> \u222b (e^x + 5sin(x)) dx<\/p>\n<p>Using the Exponential Integral:<\/p>\n<p>\u222b e^x dx = e^x + C.<\/p>\n<p>Using the Trigonometric Integral:<\/p>\n<p>\u222b sin(x) dx = -cos(x) + C.<\/p>\n<p>Since the integral of a sum is the sum of integrals:<\/p>\n<p>\u222b (e^x + 5sin(x)) dx = (e^x + C) + 5*(-cos(x) + C)<\/p>\n<p>= e^x &#8211; 5cos(x) + C.<\/p>\n<p><strong>Example 3:<\/strong> \u222b (x^3 + 2x^2 &#8211; 2x + 5) dx<\/p>\n<p>Using the Power Rule:<\/p>\n<p>\u222b (x^3) dx = (x^(3+1))\/(3+1) + C = (1\/4) * x^4 + C.<\/p>\n<p>Using the Power Rule:<\/p>\n<p>\u222b (2x^2) dx = (2 * x^(2+1))\/(2+1) + C = (2\/3) * x^3 + C.<\/p>\n<p>Using the Power Rule:<\/p>\n<p>\u222b (-2x) dx = -2 * (x^(1+1))\/(1+1) + C = -x^2 + C.<\/p>\n<p>Using the Power Rule:<\/p>\n<p>\u222b (5) dx = 5 * x + C.<\/p>\n<p>Putting it all together:<\/p>\n<p>\u222b (x^3 + 2x^2 &#8211; 2x + 5) dx = (1\/4) * x^4 + (2\/3) * x^3 &#8211; x^2 + 5x + C.<\/p>\n<p>These are just a few examples to illustrate the use of integration formulas. Remember, integration can involve more complex functions and may require multiple steps or special techniques in some cases.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Frequently_asked_questions_about_Integration_formulas\"><\/span>Frequently asked questions about Integration formulas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_integration_formula\"><\/span>What is the integration formula?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe integration formula gives a way to find the area under a curve or the accumulation of quantities. Common ones involve adding up tiny bits of areas or quantities.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_the_4_types_of_integration\"><\/span>What are the 4 types of integration?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThere are four main types: Indefinite integration, Definite integration, Line integration, and Surface integration. Each serves different mathematical purposes.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Why_is_integration_used\"><\/span>Why is integration used?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tIntegration is used to find areas under curves, volumes of shapes, or accumulate quantities. It's essential in physics for understanding concepts like work and energy.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_meaning_of_integration\"><\/span>What is the meaning of integration?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tIntegration means combining parts to form a whole. In math, it refers to summing up small pieces to determine a total or area.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_use_of_integration\"><\/span>What is the use of integration?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tIntegration helps in various fields like physics, engineering, and economics. It's used to find quantities like total distance, area under a graph, or accumulated change.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_integration\"><\/span>What is integration?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tIntegration is a mathematical process that adds up tiny parts to find a total or area. It's like piecing together small sections to get the whole picture. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the integration formula?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The integration formula gives a way to find the area under a curve or the accumulation of quantities. Common ones involve adding up tiny bits of areas or quantities.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are the 4 types of integration?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"There are four main types: Indefinite integration, Definite integration, Line integration, and Surface integration. Each serves different mathematical purposes.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Why is integration used?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Integration is used to find areas under curves, volumes of shapes, or accumulate quantities. It's essential in physics for understanding concepts like work and energy.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the meaning of integration?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Integration means combining parts to form a whole. 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