# جدول التكاملات

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## قواعد مكاملة الدوال العامة

$\int af(x)\,dx = a\int f(x)\,dx$
$\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx$
$\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int \left(d[f(x)]\int g(x)\,dx\right)\,dx$

$\int af(y)\,dy = a\int f(y)\,dy$
$\int [f(y) + g(y)]\,dy = \int f(y)\,dy + \int g(y)\,dy$
$\int f(y)g(y)\,dy = f(y)\int g(y)\,dy - \int \left(d[f(y)]\int g(y)\,dy\right)\,dy$

## تكاملات الدوال البسيطة

### الدوال غير المنطقة Irrational function

more integrals: List of integrals of irrational functions
$\int {du \over \sqrt{a^2-u^2}} = \arcsin {u \over a} + C$
$\int {-du \over \sqrt{a^2-u^2}} = \arccos {u \over a} + C$
$\int {du \over u\sqrt{u^2-a^2}} = {1 \over a}\mbox{arcsec}\,{|u| \over a} + C$

### اللوغارتيمات

more integrals: List of integrals of logarithmic functions
$\int \ln {x}\,dx = x \ln {x} - x+ C$
$\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C$

### الدوال الأسية

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

### الدوال المثلثية

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

### دوال القطع الزائد

more integrals: List of integrals of hyperbolic functions
$\int \sinh x \, dx = \cosh x + C$
$\int \cosh x \, dx = \sinh x + C$
$\int \tanh x \, dx = \ln |\cosh x| + C$
$\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C$
$\int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C$
$\int \coth x \, dx = \ln|\sinh x| + C$

## تكاملات محددة

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 definite integrals are given below. these integrals are a kind of the improper integrals

$\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi$ (see also Gamma function)
$\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi$
$\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}$ (see also Bernoulli number)
$\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}$
$\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}$
$\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)$ (where $\Gamma(z)$ is the Gamma function.)
$\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}e^\frac{b^2-4ac}{4a}$