# قوائم التكاملات

(تم التحويل من قائمة التكاملات)

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## تكاملات الدوال البسيطة

### دوال القيمة المطلقة

${\displaystyle \int \left|(ax+b)^{n}\right|\,dx={(ax+b)^{n+2} \over a(n+1)\left|ax+b\right|}+C\,\,[\,n{\text{ is odd, and }}n\neq -1\,]}$
${\displaystyle \int \left|\sin {ax}\right|\,dx={-1 \over a}\left|\sin {ax}\right|\cot {ax}+C}$
${\displaystyle \int \left|\cos {ax}\right|\,dx={1 \over a}\left|\cos {ax}\right|\tan {ax}+C}$
${\displaystyle \int \left|\tan {ax}\right|\,dx={\tan(ax)[-\ln \left|\cos {ax}\right|] \over a\left|\tan {ax}\right|}+C}$
${\displaystyle \int \left|\csc {ax}\right|\,dx={-\ln \left|\csc {ax}+\cot {ax}\right|\sin {ax} \over a\left|\sin {ax}\right|}+C}$
${\displaystyle \int \left|\sec {ax}\right|\,dx={\ln \left|\sec {ax}+\tan {ax}\right|\cos {ax} \over a\left|\cos {ax}\right|}+C}$
${\displaystyle \int \left|\cot {ax}\right|\,dx={\tan(ax)[\ln \left|\sin {ax}\right|] \over a\left|\tan {ax}\right|}+C}$

### اللوغاريتمات

more integrals: List of integrals of logarithmic functions
${\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C}$
${\displaystyle \int \log _{b}(x)\,dx=x\log _{b}(x)-x\log _{b}(e)+C}$
${\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}$

### الدوال الأسية

more integrals: List of integrals of exponential functions
${\displaystyle \int e^{x}\,dx=e^{x}+C}$
${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln(a)}}+C}$

### Trigonometric functions

more integrals: List of integrals of trigonometric functions and List of integrals of inverse trigonometric functions
${\displaystyle \int \sin {x}\,dx=-\cos {x}+C}$
${\displaystyle \int \cos {x}\,dx=\sin {x}+C}$
${\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C}$
${\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}$
${\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}$
${\displaystyle \int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C}$
${\displaystyle \int \sec ^{2}x\,dx=\tan x+C}$
${\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}$
${\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}$
${\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}$
${\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}(x-{\frac {\sin 2x}{2}})+C={\frac {1}{2}}(x-\sin x\cos x)+C}$
${\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}(x+{\frac {\sin 2x}{2}})+C={\frac {1}{2}}(x+\sin x\cos x)+C}$
${\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}$
(see integral of secant cubed)
${\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}$
${\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}$
${\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}$

### Hyperbolic functions

more integrals: List of integrals of hyperbolic functions
${\displaystyle \int \sinh x\,dx=\cosh x+C}$
${\displaystyle \int \cosh x\,dx=\sinh x+C}$
${\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}$
${\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}$
${\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C}$
${\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}$
${\displaystyle \int {\mbox{sech}}^{2}x\,dx=\tanh x+C}$

### Inverse hyperbolic functions

${\displaystyle \int \operatorname {arsinh} \,x\,dx=x\,\operatorname {arsinh} \,x-{\sqrt {x^{2}+1}}+C}$
${\displaystyle \int \operatorname {arcosh} \,x\,dx=x\,\operatorname {arcosh} \,x-{\sqrt {x^{2}-1}}+C}$
${\displaystyle \int \operatorname {artanh} \,x\,dx=x\,\operatorname {artanh} \,x+{\frac {1}{2}}\ln {(1-x^{2})}+C}$
${\displaystyle \int \operatorname {arcsch} \,x\,dx=x\,\operatorname {arcsch} \,x+\ln {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C}$
${\displaystyle \int \operatorname {arsech} \,x\,dx=x\,\operatorname {arsech} \,x-\arctan {\left({\frac {x}{x-1}}{\sqrt {\frac {1-x}{1+x}}}\right)}+C}$
${\displaystyle \int \operatorname {arcoth} \,x\,dx=x\,\operatorname {arcoth} \,x+{\frac {1}{2}}\ln {(x^{2}-1)}+C}$

### دوال خاصة

${\displaystyle \int \operatorname {Ci} (x)dx=x\,\operatorname {Ci} (x)-\sin x}$
${\displaystyle \int \operatorname {Si} (x)dx=x\,\operatorname {Si} (x)+\cos x}$
${\displaystyle \int \operatorname {Ei} (x)dx=x\,\operatorname {Ei} (x)-e^{x}}$
${\displaystyle \int \operatorname {li} (x)dx=x\,\operatorname {li} (x)-\operatorname {Ei} (2\ln x)}$
${\displaystyle \int {\frac {\operatorname {li} (x)}{x}}\,dx=\ln x\,\operatorname {li} (x)-x}$
${\displaystyle \int \operatorname {erf} (x)\,dx={\frac {e^{-x^{2}}}{\sqrt {\pi }}}+x\,{\text{erf}}(x)}$

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## 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.

${\displaystyle \int _{0}^{\infty }{{\sqrt {x}}\,e^{-x}\,dx}={\frac {1}{2}}{\sqrt {\pi }}}$  (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 ${\displaystyle e^{u}}$ , and ${\displaystyle a>0}$ )
${\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)}$  (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:

${\displaystyle \int _{a}^{b}{f(x)\,dx}=(b-a)\sum \limits _{n=1}^{\infty }{\sum \limits _{m=1}^{2^{n}-1}{\left({-1}\right)^{m+1}}}2^{-n}f(a+m\left({b-a}\right)2^{-n}).}$
${\displaystyle \int _{0}^{1}[\ln(1/x)]^{p}\,dx=p!}$

### The "sophomore's dream"

{\displaystyle {\begin{aligned}\int _{0}^{1}x^{-x}\,dx&=\sum _{n=1}^{\infty }n^{-n}&&(=1.29128599706266\dots )\\\int _{0}^{1}x^{x}\,dx&=\sum _{n=1}^{\infty }-(-1)^{n}n^{-n}&&(=0.783430510712\dots )\end{aligned}}}

منسوبة إلى يوهان برنولي.

## المصادر

• Besavilla: Engineering Review Center, Engineering Mathematics (Formulas), Mini Booklet
• I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)
• Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.)

## وصلات خارجية

### جداول التكاملات

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