{"id":9005,"date":"2025-02-01T16:44:35","date_gmt":"2025-02-01T11:14:35","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=9005"},"modified":"2025-02-01T16:44:41","modified_gmt":"2025-02-01T11:14:41","slug":"definite-integration-formulas","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/definite-integration-formulas\/","title":{"rendered":"Definite integration formulas"},"content":{"rendered":"\n

Definite integration involves finding the integral of a function over a specific interval. This process yields a number that represents the net area under the curve of the function between the two endpoints of the interval. Here are some fundamental formulas and properties related to definite integrals:<\/p>\n\n\n\n

Fundamental Theorem of Calculus<\/h3>\n\n\n\n

The Fundamental Theorem of Calculus links the concept of differentiation with integration and consists of two parts:<\/p>\n\n\n\n

    \n
  1. First Part<\/strong>: If ( f ) is continuous on ([a, b]) and ( F ) is an antiderivative of ( f ) on ([a, b]), then:
    \\[
    \\int_a^b f(x) \\, dx = F(b) – F(a)
    \\]<\/li>\n\n\n\n
  2. Second Part<\/strong>: If ( f ) is a continuous function on ([a, b]), then the function ( g ) defined by:
    \\[
    g(x) = \\int_a^x f(t) \\, dt
    \\]
    is continuous on ([a, b]), differentiable on ((a, b)), and ( g'(x) = f(x) ).<\/li>\n<\/ol>\n\n\n\n

    Properties of Definite Integrals<\/h2>\n\n\n\n

    (I) $\\int_{a}^{b} f(x) dx= \\int_{a}^{b} f(t) dt $<\/p>\n\n\n\n

    (II) $\\int_{a}^{b} f(x) dx=- \\int_{b}^{a} f(x) dx $<\/p>\n\n\n\n

    (III) $\\int_{a}^{b} f(x) dx= \\int_{a}^{c} f(x) dx + \\int_{c}^{b} f(x) dx  $<\/p>\n\n\n\n

    where a<c <b<\/p>\n\n\n\n

    (IV) $\\int_{a}^{b} f(x) dx=\\int_{a}^{b} f((a+b-x)) dx$<\/p>\n\n\n\n

    (V) $\\int_{0}^{a} f(x) dx=\\int_{0}^{a} f((a-x)) dx$<\/p>\n\n\n\n

    (VI) $\\int_{0}^{2a} f(x) dx=\\int_{0}^{a} f((x)) dx + \\int_{0}^{a} f((2a-x)) dx$<\/p>\n\n\n\n

    (VII) $\\int_{0}^{2a} f(x) dx= \\begin{cases}
    2 \\int_{0}^{a} f(x) dx & ,  f(2a-x) =f(x) \\\\
    0 &, f(2a-x) =-f(x)
    \\end{cases} $<\/p>\n\n\n\n

    (VIII) $\\int_{-a}^{a} f(x) dx= \\begin{cases}
    2 \\int_{0}^{a} f(x) dx & ,  f(x) =f(-x) \\\\
    0 &, f(x) =-f(x)
    \\end{cases} $<\/p>\n\n\n\n

    (IX) $f(x) \\geq 0$, then $\\int_{a}^{b} f(x) dx \\geq 0$<\/p>\n\n\n\n

    (X) $f(x) \\geq g(x) $, then $\\int_{a}^{b} f(x) dx \\geq \\int_{a}^{b} g(x) dx $<\/p>\n\n\n\n

    (IX) $|\\int_{a}^{b} f(x) dx| \\leq \\int_{a}^{b} |f(x)| dx$<\/p>\n\n\n\n

    Other formula’s<\/h2>\n\n\n\n

    For any real number $n \\neq -1$,
    $$
    \\int_{a}^{b} x^n \\, dx = \\frac{b^{n+1} – a^{n+1}}{n+1}
    $$<\/p>\n\n\n\n

    $$
    \\int_{a}^{b} e^x \\, dx = e^b – e^a
    $$<\/p>\n\n\n\n

    $$
    \\int_{1}^{b} \\frac{1}{x} \\, dx = \\ln(b)
    $$<\/p>\n\n\n\n

    $$
    \\int_{a}^{b} \\sin(x) \\, dx = -\\cos(b) + \\cos(a)
    $$
    $$
    \\int_{a}^{b} \\cos(x) \\, dx = \\sin(b) – \\sin(a)
    $$
    $$
    \\int_{a}^{b} \\tan(x) \\, dx = -\\ln|\\cos(b)| + \\ln|\\cos(a)|
    $$<\/p>\n\n\n\n

    A special case using polar coordinates:
    $$
    \\int_{0}^{2\\pi} \\frac{1}{2} r^2 \\, d\\theta = \\pi r^2
    $$<\/p>\n\n\n\n

    $$
    \\int_{a}^{\\infty} \\frac{1}{a^2 + x^2} \\, dx = \\frac {\\pi}{2a}
    $$<\/p>\n\n\n\n

    $$
    \\int_{a}^{\\infty} \\frac{1}{\\sqrt {a^2 – x^2}} \\, dx = \\frac {\\pi}{2}
    $$<\/p>\n\n\n\n

    $$
    \\int_{a}^{\\infty} \\sqrt {a^2 – x^2} \\, dx = \\frac {\\pi a^2}{4}
    $$<\/p>\n\n\n\n

    Example<\/strong><\/p>\n\n\n\n

    Calculate the definite integral of ( f(x) = x^3 ) from ( x = 1 ) to ( x = 2 ):<\/p>\n\n\n\n

    \\[
    \\int_1^2 x^3 \\, dx = \\left[ \\frac{x^4}{4} \\right]_1^2 = \\frac{2^4}{4} – \\frac{1^4}{4} = \\frac{16}{4} – \\frac{1}{4} = 4 – 0.25 = 3.75
    \\]<\/p>\n\n\n\n

    We hope that these Definite integration formulas will help in your preparation.<\/p>\n\n\n\n

    \n

    Other Integration Related Articles<\/p>\n\n\n\n

    Integration of tan inverse x<\/a><\/td>Integration of cos inverse x<\/a><\/td><\/tr>
    Integration of cos square x<\/a><\/td>Integration of fractional part of x<\/a><\/td><\/tr>
    integration of uv formula<\/a><\/td>integration of log x<\/a><\/td><\/tr>
    integration of sin inverse x<\/a><\/td>Integration of constant term<\/a><\/td><\/tr>
    integration of hyperbolic functions<\/a><\/td>List of Integration Formulas <\/a><\/td><\/tr>
    integration of log sinx<\/a><\/td>integration of sin x cos x dx<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"

    Definite integration involves finding the integral of a function over a specific interval. This process yields a number that represents the net area under the curve of the function between the two endpoints of the interval. Here are some fundamental formulas and properties related to definite integrals: Fundamental Theorem of Calculus The Fundamental Theorem of […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-9005","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nDefinite integration formulas - 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