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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

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

اللوغاريتمات

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

خطأ رياضيات (خطأ في الصيغة): أدخل الصيغة هنا ==تكاملات محددة ==هدا خاص بعبد النور مولاي

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.

\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 Γ(z) is the Gamma function.)
\int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}e^\frac{b^2-4ac}{4a}