{"id":5701,"date":"2023-10-16T20:01:29","date_gmt":"2023-10-16T14:31:29","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=5701"},"modified":"2024-02-04T20:17:42","modified_gmt":"2024-02-04T14:47:42","slug":"integration-formulas-list","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-formulas-list\/","title":{"rendered":"List of Integration Formulas | Basic ,Trig, Substitution, Parts, Definite | Class 12 Maths"},"content":{"rendered":"\n
\"List<\/figure>\n\n\n\n

Here is the Integration Formulas List.  The formula list is divided into below sections<\/p>\n\n\n\n

a. Basic integration formulas<\/p>\n\n\n\n

b.Integration formulas for Trigonometric Functions<\/p>\n\n\n\n

c. Integration formulas Related to Inverse Trigonometric Functions<\/p>\n\n\n\n

d. Algebra of integration<\/p>\n\n\n\n

e. Integration by Substitution<\/p>\n\n\n\n

f. Special Integrals Formula<\/p>\n\n\n\n

g. Integration by Parts<\/p>\n\n\n\n

h. Some special Integration Formulas derived using Parts method<\/p>\n\n\n\n

i. Integration of Rational algebraic functions using Partial Fractions<\/p>\n\n\n\n

j. Definite Integrals<\/p>\n\n\n\n

k. Properties of Definite Integrals<\/p>\n\n\n\n

l. Integration as Limit of Sum<\/p>\n\n\n\n

Basic Integration formulas<\/h2>\n\n\n\n

$\\int (c) = x + C$  ( Where c is a constant)<\/p>\n\n\n\n

$\\int (cx) = \\frac {cx^2}{2} + C$ ( Where c is a constant)<\/p>\n\n\n\n

$\\int (x^n) = \\frac {x^{n+1}}{n+1}$<\/p>\n\n\n\n

$\\int (e^x) = e^x + C$<\/p>\n\n\n\n

$\\int (\\frac {1}{x}) = ln |x| + c$<\/p>\n\n\n\n

$\\int (a^x) = \\frac {a^x}{ log a} + C$<\/p>\n\n\n\n

$\\int (log_{a} x) =\\frac {1}{x ln a} + C$<\/p>\n\n\n\n

Integration formulas for Trigonometric Functions<\/h2>\n\n\n\n

$\\int (\\cos x) = \\sin x + C$<\/p>\n\n\n\n

$\\int (\\sin x) = – \\cos x + C$<\/p>\n\n\n\n

$\\int ( \\sec^2 x) = \\tan x + C$<\/p>\n\n\n\n

$\\int (\\csc^2 x) = -\\cot x + C$<\/p>\n\n\n\n

$\\int ( \\sec (x) \\tan (x) )=\\sec x + C$<\/p>\n\n\n\n

$\\int (\\csc (x) \\cot (x)) = -\\csc x + C$<\/p>\n\n\n\n

$ \\int (\\tan x) = ln |\\sec x| + C$<\/p>\n\n\n\n

$ \\int (\\cot x) = ln |\\sin x| + C$<\/p>\n\n\n\n

$ \\int (\\sec x) = ln |\\sec x + \\tan x| + C$<\/p>\n\n\n\n

$ \\int (\\csc x) = ln |\\csc x – \\cot x| + C$<\/p>\n\n\n\n

Integration formulas Related to Inverse Trigonometric Functions<\/h2>\n\n\n\n

$\\int ( \\frac {1}{\\sqrt {1-x^2} } ) = \\sin^{-1}x + C$<\/p>\n\n\n\n

$\\int (\\frac {1}{\\sqrt {1-x^2}}) = – \\cos ^{-1}x  +C$<\/p>\n\n\n\n

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

$\\int ( \\frac {1}{1 + x^2}) = -\\cot ^{-1}x + C$<\/p>\n\n\n\n

$\\int (\\frac {1}{|x|\\sqrt {x^-1}}) = -sec^{-1} x + C $<\/p>\n\n\n\n

$\\int (\\frac {1}{|x|\\sqrt {x^-1}}) = -cosec^{-1} x + C $<\/p>\n\n\n\n

Algebra of integration<\/h2>\n\n\n\n

Multiplication by Constant<\/p>\n\n\n\n

$\\int [cf(x)] dx = c \\int f(x) dx $<\/p>\n\n\n\n

Example<\/em><\/p>\n\n\n\n

$\\int [2x ] dx = 2 \\int (x) dx =x^2 + C$<\/p>\n\n\n\n

Addition and Subtraction<\/em><\/p>\n\n\n\n

$\\int [f(x)+g(x)] dx=\\int f(x)  dx+ \\int g(x) dx$<\/p>\n\n\n\n

$\\int [f(x)-g(x)]dx=\\int f(x) dx – \\int  g(x) dx$<\/p>\n\n\n\n

Example<\/em><\/p>\n\n\n\n

$\\int [sinx -cos x ] dx = \\int sin x  dx- \\int cos x dx=-cos (x) – sin(x) + C$<\/p>\n\n\n\n

Integration by Substitution<\/h2>\n\n\n\n

A. if $ \\int f(x) dx =  g(x) $ then $\\int f(ax+ b) = \\frac {1}{a} g(x) $<\/p>\n\n\n\n

$\\int (ax+b)^n = \\frac {1}{a} \\frac {(ax+ b)^{n+1}}{n+1} + C$<\/p>\n\n\n\n

$\\int e^{ax+b} =\\frac {1}{a}  e^{ax+b} + C$<\/p>\n\n\n\n

$\\int (\\frac {1}{ax+b}) = \\frac {1}{a} ln |ax +b| + c$<\/p>\n\n\n\n

$\\int a^{bx+c} = \\frac {1}{b} \\frac {a^{bx+c}}{ log a} + C$<\/p>\n\n\n\n

$\\int \\cos (ax+b) = \\frac {1}{a}  \\sin (ax+b) + C$<\/p>\n\n\n\n

$\\int \\sin (ax+b) = – \\frac {1}{a} \\cos (ax+b) + C$<\/p>\n\n\n\n

$\\int \\sec^2 (ax+b) = \\frac {1}{a}  \\tan (ax +b) + C$<\/p>\n\n\n\n

$\\int \\csc^2 (ax+b) = – \\frac {1}{a}  \\cot (ax+b)+ C$<\/p>\n\n\n\n

$ \\int \\tan (ax+b) =- \\frac {1}{a}  ln |\\cos (ax+b)| + C$<\/p>\n\n\n\n

$ \\int \\cot (ax+b) = \\frac {1}{a}  ln |\\sin (ax+b)| + C$<\/p>\n\n\n\n

$ \\int \\sec (ax+b) =\\frac {1}{a} ln |\\sec (ax+b) + \\tan (ax+b)| + C$<\/p>\n\n\n\n

$ \\int \\csc (ax+b) = \\frac {1}{a} ln |\\csc (ax+b) – \\cot (ax+b)| + C$<\/p>\n\n\n\n

B.  $\\int \\frac {f^{‘} (x)}{f(x)} dx  = ln | f(x)| + C$<\/p>\n\n\n\n

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

$\\int \\frac {1}{1 + e^{-x}} dx = \\int \\frac {1}{1 + 1\/e^x} dx = \\int \\frac {e^x}{1+ e^x} dx$<\/p>\n\n\n\n

Now this above form<\/p>\n\n\n\n

$= ln |1 + e^x| + C$<\/p>\n\n\n\n

C.  $\\int  [f(x)]^n f^{‘} x dx = \\frac { [f(x)]^{n+1}}{n +1 }  , n \\ne -1 $<\/p>\n\n\n\n

Special Integrals Formula<\/h2>\n\n\n\n

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

$\\int \\frac {1}{x^2 – a^2} dx = \\frac {1}{2a} ln  |\\frac {x-a}{x+a}| + C$<\/p>\n\n\n\n

$\\int \\frac {1}{a^2 – x^2} dx = \\frac {1}{2a} ln  |\\frac {a+x}{a-x}| + C$<\/p>\n\n\n\n

$\\int \\frac {1}{\\sqrt {a^2 – x^2}} dx =  \\sin ^{-1} (\\frac {x}{a}) + C$<\/p>\n\n\n\n

$\\int \\frac {1}{\\sqrt {a^2  + x^2}} dx =  ln |x + \\sqrt {a^2  + x^2}|   + C$<\/p>\n\n\n\n

$\\int \\frac {1}{\\sqrt {x^2  – a^2}} dx =  ln |x + \\sqrt {x^2  – a^2}|   + C$<\/p>\n\n\n\n

Integration by Parts<\/h2>\n\n\n\n

A. $\\int f(x) g(x) dx= f(x)  (\\int g(x) dx ) – \\int \\left \\{ \\frac {df(x)}{dx} \\int g(x) dx \\right \\}dx$<\/p>\n\n\n\n

We can decide first function using the word ILATE<\/p>\n\n\n\n

I -> Inverse trigonometric functions<\/p>\n\n\n\n

L -> Logarithmic functions<\/p>\n\n\n\n

A-> Algebraic functions<\/p>\n\n\n\n

T -> trigonometric functions<\/p>\n\n\n\n

E -> Exponential functions<\/p>\n\n\n\n

B. $\\int e^x{ f(x) + f^{‘} (x) } dx =  e^x f(x)  + C$<\/p>\n\n\n\n

Some special Integration Formulas derived using Parts method<\/h2>\n\n\n\n

$ \\int \\sqrt {a^2 – x^2} dx = \\frac {1}{2} x \\sqrt {a^2 – x^2} + \\frac {1}{2} a^2 \\sin^{-1} \\frac {x}{a} + C$<\/p>\n\n\n\n

$ \\int \\sqrt {a^2 + x^2} dx = \\frac {1}{2} x \\sqrt {a^2 + x^2} + \\frac {1}{2} a^2  ln |x +\\sqrt {a^2 + x^2}|  + C$<\/p>\n\n\n\n

$ \\int \\sqrt {x^2 -a ^2} dx = \\frac {1}{2} x \\sqrt {x^2 – a^2} – \\frac {1}{2} a^2  ln |x +\\sqrt {x^2 – a^2}|  + C$<\/p>\n\n\n\n

The above formula can be to use to integrate the below type of function<\/p>\n\n\n\n

$ \\int \\sqrt {ax^2 + bx + c} dx$<\/p>\n\n\n\n

We can convert $ax^2 + bx + c$ into above using square method<\/p>\n\n\n\n

$ \\int  (px +q) \\sqrt {ax^2 + bx + c} dx$<\/p>\n\n\n\n

We can express $px + q = \\lambda \\frac {d}{dx} (ax^2 + bx +c) + \\mu$<\/p>\n\n\n\n

We find the values of $ \\lambda$ and $\\mu$<\/p>\n\n\n\n

Now this will get converted into entities. One of the integration can be obtained from above formula and one from<\/p>\n\n\n\n

$\\int  [f(x)]^n f^{‘} x dx = \\frac { [f(x)]^{n+1}}{n +1 }  , n \\ne -1 $<\/p>\n\n\n\n

Integration of Rational algebraic functions using Partial Fractions<\/h2>\n\n\n\n

$ \\int \\frac {px +q}{(x-a)(x-b)} dx =\\int  \\left \\{  \\frac {A}{x-a} + \\frac {B}{x-b}   \\right \\} dx$<\/p>\n\n\n\n

$ \\int \\frac {px^2 +qx + r}{(x-a)(x-b)(x-c)} dx =\\int  \\left \\{  \\frac {A}{x-a} + \\frac {B}{x-b}  + \\frac {C}{x-c}  \\right \\} dx$<\/p>\n\n\n\n

$ \\int \\frac {px +q}{(x-a)^2} dx =\\int  \\left \\{  \\frac {A}{x-a} + \\frac {B}{(x-a)^2}   \\right \\} dx$<\/p>\n\n\n\n

$ \\int \\frac {px^2 +qx + r}{(x-a)^2(x-c)} dx =\\int  \\left \\{  \\frac {A}{x-a} + \\frac {B}{(x-a)^2}  + \\frac {C}{x-c}  \\right \\} dx$<\/p>\n\n\n\n

$ \\int \\frac {px^2 +q+r}{(x-a)(x^2 + bx +c)} dx =\\int  \\left \\{  \\frac {A}{x-a} + \\frac {Bx +C}{x^2 + bx +c}   \\right \\} dx$<\/p>\n\n\n\n

where $x^2 + bx +c$ is a irreducible quadratic<\/p>\n\n\n\n

\n

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

Integration of modulus function<\/a><\/td>Integration of tan x<\/a><\/td><\/td><\/td><\/tr>
Integration of sec x<\/a><\/td>Integration of greatest integer function<\/a><\/td><\/td><\/td><\/tr>
Integration of trigonometric functions<\/a><\/td>Integration of cot x<\/a><\/td><\/td><\/td><\/tr>
Integration of cosec x<\/a><\/td>Integration of sinx<\/a><\/td><\/td><\/td><\/tr>
Integration of irrational functions<\/a><\/td>integration of root x<\/a><\/td><\/td><\/td><\/tr>
Integration of fractional part of x<\/a><\/td>Integration of cos square x<\/a><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n\n\n\n

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

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

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

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

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

$\\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

if (x) is a continuous function defined on [0,a],then<\/p>\n\n\n\n

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

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

$\\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

Integration as Limit of Sum<\/h2>\n\n\n\n

$\\int_{a}^{b} f(x) dx = \\lim_{h \\rightarrow 0} h[f(a) + f(a+h) + f(a+2h)+…..+f(a + (n-1)h)]$<\/p>\n\n\n\n

where $h= \\frac {b-a}{n}$<\/p>\n\n\n\n

\n\n\n\n

Related Articles<\/strong><\/p>\n\n\n\n

Inverse Trigonometric Function Formulas<\/a>
Differentiation formulas<\/a>
Trigonometry Formulas for class 11<\/a>
Physics formulas pdf<\/a><\/p>\n\n\n\n

\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here is the Integration Formulas List.  The formula list is divided into below sections a. Basic integration formulas b.Integration formulas for Trigonometric Functions c. Integration formulas Related to Inverse Trigonometric Functions d. Algebra of integration e. Integration by Substitution f. Special Integrals Formula g. Integration by Parts h. Some special Integration Formulas derived using Parts […]<\/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":[506],"class_list":["post-5701","post","type-post","status-publish","format-standard","hentry","category-maths","tag-integration"],"yoast_head":"\nIntegration Formulas List for Class 12 - 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Basic integration formulas b.Integration formulas for Trigonometric Functions c. Integration formulas Related to Inverse Trigonometric Functions d. Algebra of integration e. Integration by Substitution f. Special Integrals Formula g. Integration by Parts h. Some special Integration Formulas derived using Parts…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5701","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=5701"}],"version-history":[{"count":6,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5701\/revisions"}],"predecessor-version":[{"id":9007,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5701\/revisions\/9007"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=5701"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=5701"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=5701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}