समाकल सूची

समाकलन, कलन की दो प्रमुख क्रियाओं में से एक है। अवकलन इस दृष्टि से समाकलन से भिन्न है कि अवकलज निकालने के लिये छोटे-छोटे और सरल नियम व उपाय हैं; जिनकी सहायता से कठिन से कठिन फलनों का भी अवकलज निकाला जा सकता है। समाकलन इस दृष्टि से कठिन है। इसलिये ज्ञात समाकलनों की सूची बहुत उपयोगी होती है।

नीचे कुछ अति सामान्य फलनों के समाकल दिये गये हैं:(x)

(ये विधियाँ तभी लागू होंगी यदि दिया हुआ फलन समाकलनीय हो)

${\displaystyle \int af(x)\,dx=a\int f(x)\,dx\qquad {\mbox{(}a\neq 0{\mbox{, constant)}\,\!}$

${\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}$

${\displaystyle \int f'(x)g(x)\,dx=f(x)g(x)-\int f(x)g'(x)\,dx}$ -- ( खंडश: समाकलन (इण्टीग्रेशन बाई पार्ट्स) )

${\displaystyle \int {f'(x) \over f(x)}\,dx=\ln {\left|f(x)\right|}+C}$

${\displaystyle \int {f'(x)f(x)}\,dx={1 \over 2}[f(x)]^{2}+C}$

${\displaystyle \int [f(x)]^{n}f'(x)\,dx={[f(x)]^{n+1} \over n+1}+C\qquad {\mbox{(for }n\neq -1{\mbox{)}\,\!}$

${\displaystyle \int \,{\rm {d}x=x+C}$

${\displaystyle \int x^{n}\,{\rm {d}x={\frac {x^{n+1}{n+1}+C\qquad {\mbox{ if }n\neq -1}$

${\displaystyle \int {dx \over x}=\ln {\left|x\right|}+C}$

${\displaystyle \int {dx \over {a^{2}+b^{2}x^{2}={1 \over ab}\arctan {dx \over a}+C}$

${\displaystyle \int {dx \over {\sqrt {a^{2}-x^{2}=\sin ^{-1}{x \over a}+C}$

${\displaystyle \int {-dx \over {\sqrt {a^{2}-x^{2}=\cos ^{-1}{x \over a}+C}$

${\displaystyle \int {dx \over x{\sqrt {x^{2}-a^{2}={1 \over a}\sec ^{-1}{|x| \over a}+C}$

${\displaystyle \int \ln {x}\,dx=x\ln {x}-x+C}$

${\displaystyle \int \log _{b}{x}\,dx=x\log _{b}{x}-x\log _{b}{e}+C}$

${\displaystyle \int e^{x}\,dx=e^{x}+C}$

${\displaystyle \int a^{x}\,dx={\frac {a^{x}{\ln {a}+C}$

${\displaystyle \int \sin {x}\,dx=-\cos {x}+C}$

${\displaystyle \int \cos {x}\,dx=\sin {x}+C}$

${\displaystyle \int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C}$

${\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}$

${\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}$

${\displaystyle \int {\mbox{cosec }{x}\,dx=\ln {\left|{\mbox{cosec }{x}-{\mbox{cot }{x}\right|}+C}$

${\displaystyle \int \sec ^{2}x\,dx=\tan x+C}$

${\displaystyle \int {\mbox{cosec }{x}{\mbox{cot }{x}\,dx=-{\mbox{cosec }{x}+C}$

${\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}$

${\displaystyle \int {\mbox{cosec}^{2}x\,dx=-{\mbox{cot }x+C}$

${\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}(x-\sin x\cos x)+C}$

${\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}(x+\sin x\cos x)+C}$

${\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}\sec x\tan x+{\frac {1}{2}\ln |\sec x+\tan x|+C}$

(see integral of secant cubed)

${\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}{n}+{\frac {n-1}{n}\int \sin ^{n-2}{x}\,dx}$

${\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}{n}+{\frac {n-1}{n}\int \cos ^{n-2}{x}\,dx}$

${\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}\ln {\left|1+x^{2}\right|}+C}$

${\displaystyle \int \sinh x\,dx=\cosh x+C}$

${\displaystyle \int \cosh x\,dx=\sinh x+C}$

${\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}$

${\displaystyle \int {\mbox{csch}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}$

${\displaystyle \int {\mbox{sech}\,x\,dx=\arctan(\sinh x)+C}$

${\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}$

${\displaystyle \int {\mbox{sech}^{2}x\,dx=\tanh x+C}$

${\displaystyle \int \operatorname {arcsinh} x\,dx=x\operatorname {arcsinh} x-{\sqrt {x^{2}+1}+C}$

${\displaystyle \int \operatorname {arccosh} x\,dx=x\operatorname {arccosh} x-{\sqrt {x^{2}-1}+C}$

${\displaystyle \int \operatorname {arctanh} x\,dx=x\operatorname {arctanh} x+{\frac {1}{2}\log {(1-x^{2})}+C}$

${\displaystyle \int \operatorname {arccsch} \,x\,dx=x\operatorname {arccsch} x+\log {\left[x\left({\sqrt {1+{\frac {1}{x^{2}+1\right)\right]}+C}$

${\displaystyle \int \operatorname {arcsech} \,x\,dx=x\operatorname {arcsech} x-\arctan {\left({\frac {x}{x-1}{\sqrt {\frac {1-x}{1+x}\right)}+C}$

${\displaystyle \int \operatorname {arccoth} \,x\,dx=x\operatorname {arccoth} x+{\frac {1}{2}\log {(x^{2}-1)}+C}$

${\displaystyle \int {\frac {1}{(x^{2}+1)^{n}\,\mathrm {d} x={\frac {1}{2n-2}\cdot {\frac {x}{(x^{2}+1)^{n-1}+{\frac {2n-3}{2n-2}\cdot \int {\frac {1}{(x^{2}+1)^{n-1}\,\mathrm {d} x,\quad n\geq 2}$

${\displaystyle \int \sin ^{n}(x)dx={\frac {n-1}{n}\int \sin ^{n-2}(x)dx-{\frac {1}{n}\cos(x)\sin ^{n-1}(x),\quad n\geq 2}$

${\displaystyle \int \cos ^{n}(x)dx={\frac {n-1}{n}\int \cos ^{n-2}(x)dx+{\frac {1}{n}\sin(x)\cos ^{n-1}(x),\quad n\geq 2}$

There are some functions whose antiderivatives cannot be expressed in closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.

${\displaystyle \int _{0}^{\infty }{\sqrt {x}\,e^{-x}\,dx}={\frac {1}{2}{\sqrt {\pi }$ (see also Gamma function)

${\displaystyle \int _{0}^{\infty }{e^{-x^{2}\,dx}={\frac {1}{2}{\sqrt {\pi }$ (the Gaussian integral)

${\displaystyle \int _{0}^{\infty }{\frac {x}{e^{x}-1}\,dx}={\frac {\pi ^{2}{6}$ (see also Bernoulli number)

${\displaystyle \int _{0}^{\infty }{\frac {x^{3}{e^{x}-1}\,dx}={\frac {\pi ^{4}{15}$

${\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}\,dx={\frac {\pi }{2}$

${\displaystyle \int _{0}^{\frac {\pi }{2}\sin ^{n}{x}\,dx=\int _{0}^{\frac {\pi }{2}\cos ^{n}{x}\,dx={\frac {1\cdot 3\cdot 5\cdot \cdots \cdot (n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot n}{\frac {\pi }{2}$ (if n is an even integer and ${\displaystyle \scriptstyle {n\geq 2}$)

${\displaystyle \int _{0}^{\frac {\pi }{2}\sin ^{n}{x}\,dx=\int _{0}^{\frac {\pi }{2}\cos ^{n}{x}\,dx={\frac {2\cdot 4\cdot 6\cdot \cdots \cdot (n-1)}{3\cdot 5\cdot 7\cdot \cdots \cdot n}$ (if ${\displaystyle \scriptstyle {n}$ is an odd integer and ${\displaystyle \scriptstyle {n\geq 3}$)

${\displaystyle \int _{0}^{\infty }{\frac {\sin ^{2}{x}{x^{2}\,dx={\frac {\pi }{2}$

${\displaystyle \int _{0}^{\infty }x^{z-1}\,e^{-x}\,dx=\Gamma (z)}$ (where ${\displaystyle \Gamma (z)}$ is the Gamma function)

${\displaystyle \int _{-\infty }^{\infty }e^{-(ax^{2}+bx+c)}\,dx={\sqrt {\frac {\pi }{a}\exp \left[{\frac {b^{2}-4ac}{4a}\right]}$ (where ${\displaystyle \exp[u]}$ is the exponential function ${\displaystyle e^{u}$.)

${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)}$ (where ${\displaystyle I_{0}(x)}$ is the modified Bessel function of the first kind)

${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}\right)}$

${\displaystyle \int _{-\infty }^{\infty }{(1+x^{2}/\nu )^{-(\nu +1)/2}dx}={\frac {\sqrt {\nu \pi }\ \Gamma (\nu /2)}{\Gamma ((\nu +1)/2))}\,}$ (${\displaystyle \nu >0\,}$, this is related to the probability density function of the Student's t-distribution)

The method of exhaustion provides a formula for the general case when no antiderivative exists:

${\displaystyle \int _{a}^{b}{f(x)\,dx}=(b-a)\sum \limits _{n=1}^{\infty }{\sum \limits _{m=1}^{2^{n}-1}{\left({-1}\right)^{m+1}2^{-n}f(a+m\left({b-a}\right)2^{-n})}$

${\displaystyle {\begin{aligned}\int _{0}^{1}x^{-x}\,dx&=\sum _{n=1}^{\infty }n^{-n}&&(=1.291285997\dots )\\\int _{0}^{1}x^{x}\,dx&=\sum _{n=1}^{\infty }-(-1)^{n}n^{-n}&&(=0.783430510712\dots )\end{aligned}$

(जॉन बर्नौली के नाम से प्रसिद्ध; see sophomore's dream).

• गामा फलन: ${\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}\,e^{-x}\,dx}$

• एरर फलन: ${\displaystyle {\text{erf}(x)={\frac {2}{\sqrt {\pi }\int _{0}^{x}e^{-t^{2}dt}$

• गघुगणकीय समाकल: ${\displaystyle {\text{Li}(x)=\int _{0}^{x}{\frac {dx}{\ln x}$

• एलिप्टिक समाकल : ${\displaystyle F(a,\theta )=\int _{0}^{\text{sin }\theta }{\frac {dx}{\sqrt {(1-x^{2})(1-a^{2}x^{2})}$

• ज्या समाकल: ${\displaystyle {\text{Si}(x)=\int _{0}^{x}{\frac {\text{sin }t}{t}dt}$

• कोज्या समाकल: ${\displaystyle {\text{Ci}(x)=-\int _{x}^{\infty }{\frac {\text{cos}t}{t}dt}$

• परिमेय फलनों के समाकल की सूची (List of Integrands of Rational functions)

• अवकलजों की सूची

• गणितीय सर्वसमिकाओं की सूची

• S.O.S. Mathematics: Tables and Formulas (warning: may serve popunders)
• Paul's Online Math Notes
• A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals
• Math Major: A Table of Integrals
• O'Brien, Francis J. Jr. 500 Integrals Derived integrals of exponential and logarithmic functions
• Rule-based Mathematics Precisely defined indefinite integration rules covering a wide class of integrands

• V. H. Moll, The Integrals in Gradshteyn and Ryzhik

• Integration examples for Wolfram Alpha

• wxmaxima gui for Symbolic and numeric resolution of many mathematical problems