{"id":8919,"date":"2024-01-28T03:42:27","date_gmt":"2024-01-27T22:12:27","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8919"},"modified":"2024-01-28T03:42:32","modified_gmt":"2024-01-27T22:12:32","slug":"integration-of-sec-square-x","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-sec-square-x\/","title":{"rendered":"integration of sec square x"},"content":{"rendered":"\n

The integral of the sec square function, $\\sec ^2x$, can be found using basic calculus principles. The integral of ($\\sec ^2x$) with respect to (x) is:<\/p>\n\n\n\n

\\[
\\int \\sec^2 x \\, dx = \\tan x + C
\\]<\/p>\n\n\n\n

Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up to an arbitrary constant. This result is derived from the fact that the derivative of ($\\tan x$) is $\\sec^2 x$.<\/p>\n\n\n\n

Integration of sec square x by Principal of Derivatives<\/h2>\n\n\n\n

we can use the fundamental theorem of calculus, which states that if a function f is the derivative of another function F, then the integral of f is F plus a constant.<\/p>\n\n\n\n

Now we know that<\/p>\n\n\n\n

$\\frac {d}{dx} \\tan (x)= \\sec^2(x)$
or
$\\frac {d}{dx} (\\tan(x)+ C) = \\sec^2(x)$<\/p>\n\n\n\n

Where C is the constant<\/p>\n\n\n\n

Now from fundamental theorem of calculus stated above, we can say that<\/p>\n\n\n\n

\\[
\\int \\sec^2 x \\, dx = \\tan x + C
\\]<\/p>\n\n\n\n

Integration of sec square x by substitution method<\/h2>\n\n\n\n

Let $t= \\tan x$
$dt = \\sec^2 x dx$
Therefore<\/p>\n\n\n\n

\\[
\\int \\sec^2 x \\, dx = \\int \\; dt
\\]<\/p>\n\n\n\n

$= t + C$<\/p>\n\n\n\n

Substituting back the values<\/p>\n\n\n\n

\\[
\\int \\sec^2 x \\, dx = \\tan x + C
\\]<\/p>\n\n\n\n

Solved Examples based on Integration of sec square x<\/h2>\n\n\n\n

Example 1<\/strong><\/p>\n\n\n\n

$\\int x \\sec^2 x \\, dx$<\/p>\n\n\n\n

Solution<\/strong><\/p>\n\n\n\n

To solve the integral , we need to use integration by parts, which is given by the formula:<\/p>\n\n\n\n

$$
\\int u \\, dv = uv – \\int v \\, du
$$<\/p>\n\n\n\n

Here, we can choose $ u = x $ and $ dv = \\sec^2 x \\, dx $. Then we need to find $ du $ and $ v $.<\/p>\n\n\n\n

    \n
  1. $ u = x $ implies $ du = dx $.<\/li>\n\n\n\n
  2. $ dv = \\sec^2 x \\, dx $ implies $ v = \\int \\sec^2 x \\, dx $.<\/li>\n<\/ol>\n\n\n\n

    The integral of $\\sec^2 x$ is $ \\tan x $, so $ v = \\tan x $.<\/p>\n\n\n\n

    Substituting these<\/p>\n\n\n\n

    $$
    \\int x \\sec^2 x \\, dx = x(\\tan x) – \\int (\\tan x) \\, dx
    $$<\/p>\n\n\n\n

    Now we know that $\\int \\tan x \\, dx= \\ln |sec(x)|$<\/p>\n\n\n\n

    Therefore, the final value is
    $$
    =x \\tan x + \\log|\\sec x| + C
    $$<\/p>\n\n\n\n

    where $C$ is the constant of integration.<\/p>\n\n\n\n

    Example 2<\/strong><\/p>\n\n\n\n

    $\\int \\sec^2 (2x) \\, dx$<\/p>\n\n\n\n

    Solution<\/strong><\/p>\n\n\n\n

    Let $t= 2 x$
    $dt = 2 dx$
    Therefore<\/p>\n\n\n\n

    $\\int \\sec^2 (2x) \\, dx = \\frac {1}{2} \\int \\sec^2 (t) \\, dt $
    $= \\frac {1}{2}( \\tan t + C)$<\/p>\n\n\n\n

    Substituting back<\/p>\n\n\n\n

    $\\int \\sec^2 (2x) \\, dx = \\frac {1}{2} ( \\tan 2x) + C)$<\/p>\n\n\n\n

    I hope you like article on Integration of sec square x. <\/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":"

    The integral of the sec square function, $\\sec ^2x$, can be found using basic calculus principles. The integral of ($\\sec ^2x$) with respect to (x) is: \\[\\int \\sec^2 x \\, dx = \\tan x + C\\] Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up […]<\/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-8919","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nintegration of sec square x - physicscatalyst's Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/integration-of-sec-square-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"integration of sec square x - physicscatalyst's Blog\" \/>\n<meta property=\"og:description\" content=\"The integral of the sec square function, $sec ^2x$, can be found using basic calculus principles. 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