समाकल सूची

साँचा:sidebar with collapsible lists 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)

फलनों के समाकलन की सामान्य विधियाँ

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

<math>\int af(x)\,dx = a\int f(x)\,dx \qquad\mbox{(}a \neq 0 \mbox{, constant)}\,\!</math>

<math>\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx</math>

<math>\int f'(x)g(x)\,dx = f(x)g(x) - \int f(x)g'(x)\,dx</math> -- ( खंडश: समाकलन (इण्टीग्रेशन बाई पार्ट्स) )

<math>\int {f'(x)\over f(x)}\,dx= \ln{\left|f(x)\right|} + C </math>

<math>\int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + C </math>

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

सरल फलनों के समाकल

परिमेय फलन

<math>\int \,{\rm d}x = x + C</math>

<math>\int x^n\,{\rm d}x = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ if }n \ne -1</math>

<math>\int {dx \over x} = \ln{\left|x\right|} + C</math>

<math>\int {dx \over {a^2+b^2x^2}} = {1 \over ab}\arctan {dx \over a} + C</math>

अपरिमेय फलन

<math>\int {dx \over \sqrt{a^2-x^2}} = \sin^{-1} {x \over a} + C</math>

<math>\int {-dx \over \sqrt{a^2-x^2}} = \cos^{-1} {x \over a} + C</math>

<math>\int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \sec^{-1} {|x| \over a} + C</math>

लघुगणकीय फलन

<math>\int \ln {x}\,dx = x \ln {x} - x + C</math>

<math>\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C</math>

चरघातांकी फलन

<math>\int e^x\,dx = e^x + C</math>

<math>\int a^x\,dx = \frac{a^x}{\ln{a}} + C</math>

त्रिकोणमित्तीय फलन

<math>\int \sin{x}\, dx = -\cos{x} + C</math>

<math>\int \cos{x}\, dx = \sin{x} + C</math>

<math>\int \tan{x} \, dx = \ln{\left| \sec {x} \right|} + C</math>

<math>\int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C</math>

<math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C</math>

<math>\int \mbox{cosec }{x} \, dx = \ln{\left| \mbox{cosec }{x} - \mbox{cot }{x}\right|} + C</math>

<math>\int \sec^2 x \, dx = \tan x + C</math>

<math>\int \mbox{cosec }{x} \mbox{cot }{x} \, dx = -\mbox{cosec }{x} + C</math>

<math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + C</math>

<math>\int \mbox{cosec}^2 x \, dx = -\mbox{cot }x + C</math>

<math>\int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + C</math>

<math>\int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C</math>

<math>\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C</math>

(see integral of secant cubed)

<math>\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx</math>

<math>\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx</math>

<math>\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C</math>

हाइपरबोलिक फलन

<math>\int \sinh x \, dx = \cosh x + C</math>

<math>\int \cosh x \, dx = \sinh x + C</math>

<math>\int \tanh x \, dx = \ln| \cosh x | + C</math>

<math>\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C</math>

<math>\int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C</math>

<math>\int \coth x \, dx = \ln| \sinh x | + C</math>

<math>\int \mbox{sech}^2 x\, dx = \tanh x + C</math>

इन्वर्स हाइपरबोलिक फलन

<math>\int \operatorname{arcsinh} x \, dx = x \operatorname{arcsinh} x - \sqrt{x^2+1} + C</math>

<math>\int \operatorname{arccosh} x \, dx = x \operatorname{arccosh} x - \sqrt{x^2-1} + C</math>

<math>\int \operatorname{arctanh} x \, dx = x \operatorname{arctanh} x + \frac{1}{2}\log{(1-x^2)} + C</math>

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

<math>\int \operatorname{arcsech}\,x \, dx = x \operatorname{arcsech} x- \arctan{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + C</math>

<math>\int \operatorname{arccoth}\,x \, dx = x \operatorname{arccoth} x+ \frac{1}{2}\log{(x^2-1)} + C</math>

प्रतिवर्तन सूत्र (रिकर्सन फॉर्मूले)

<math>\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</math>

<math>\int\sin^n(x)d x =

\frac{n-1}{n}\int\sin^{n-2}(x)dx -\frac{1}{n}\cos(x)\sin^{n-1}(x),\quad n\geq 2</math>

<math>\int\cos^n(x)d x =

\frac{n-1}{n}\int\cos^{n-2}(x)dx +\frac{1}{n}\sin(x)\cos^{n-1}(x),\quad n\geq 2</math>

Definite integrals lacking closed-form antiderivatives

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.

<math>\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi</math> (see also Gamma function)

<math>\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi</math> (the Gaussian integral)

<math>\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}</math> (see also Bernoulli number)

<math>\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}</math>

<math>\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}</math>

<math>\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}</math> (if n is an even integer and <math> \scriptstyle{n \ge 2}</math>)

<math>\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}</math> (if <math> \scriptstyle{n} </math> is an odd integer and <math> \scriptstyle{n \ge 3} </math>)

<math>\int_0^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2}</math>

<math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)</math> (where <math>\Gamma(z)</math> is the Gamma function)

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

<math>\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)</math> (where <math>I_{0}(x)</math> is the modified Bessel function of the first kind)

<math>\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left(\sqrt{x^2 + y^2}\right) </math>

<math>\int_{-\infty}^{\infty}{(1 + x^2/\nu)^{-(\nu + 1)/2}dx} = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2))}\,</math> (<math>\nu > 0\,</math>, 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:

<math>\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})</math>

The "sophomore's dream"

<math>\begin{align}

\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)^nn^{-n} &&(= 0.783430510712\dots) \end{align}</math>

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

विशेष फलन

• गामा फलन: <math>\Gamma(z) = \int_0^\infty x^{z-1}\,e^{-x}\,dx</math>

• एरर फलन: <math>\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt </math>

• गघुगणकीय समाकल: <math>\text{Li}(x) = \int_0^x \frac{dx}{\ln x}</math>

• एलिप्टिक समाकल : <math>F(a,\theta) = \int_0^{\text{sin }\theta} \frac{dx}{\sqrt{(1 - x^2)(1 - a^2 x^2)}}</math>

• ज्या समाकल: <math>\text{Si}(x) = \int_0^x \frac{\text{sin }t}{t}dt</math>

• कोज्या समाकल: <math>\text{Ci}(x) = - \int_x^\infty \frac{\text{cos} t}{t}dt</math>

इन्हें भी देखें

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

• अवकलजों की सूची

• गणितीय सर्वसमिकाओं की सूची

बाहरी कड़ियाँ

समाकलजों 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

उपपत्तियाँ

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

आनलाइन सेवाएँ

• Integration examples for Wolfram Alpha

मुक्त स्रोत प्रोग्राम

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