WikiSysop: ١ مراجعة: الصفحات في تصنيف رياضيات

2010-11-12T21:16:16Z

١ مراجعة: الصفحات في تصنيف رياضيات

صفحة جديدة

==قواعد مكاملة الدوال العامة==

:<math>\int af(x)\,dx = a\int f(x)\,dx</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)\int g(x)\,dx - \int \left(d[f(x)]\int g(x)\,dx\right)\,dx</math>

== تكاملات الدوال البسيطة ==

=== [[الدوال غير المنطقة]] Irrational function ===
:''more integrals: [[List of integrals of irrational functions]]''
:<math>\int {du \over \sqrt{a^2-u^2}} = \arcsin {u \over a} + C</math>
:<math>\int {-du \over \sqrt{a^2-u^2}} = \arccos {u \over a} + C</math>
:<math>\int {du \over u\sqrt{u^2-a^2}} = {1 \over a}\mbox{arcsec}\,{|u| \over a} + C</math>

=== [[اللوغاريتمات]] ===
:''more integrals: [[List of integrals of logarithmic functions]]''
:<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>

=== [[الدوال الأسية]] ===
:''more integrals: [[List of integrals of exponential functions]]''
:<math>\int e^x\,dx = e^x + C</math>
:<math>\int a^x\,dx = \frac{a^x}{\ln{a}} + C</math>

=== [[دوال مثلثية|الدوال المثلثية]] ===
:''more integrals: [[List of integrals of trigonometric functions]] and [[List of integrals of arc functions]]''
:<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| \cos {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 \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C</math>
:<math>\int \sec^2 x \, dx = \tan x + C</math>
:<math>\int \csc^2 x \, dx = -\cot x + C</math>
:<math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + C</math>
:<math>\int \csc{x} \, \cot{x} \, dx = - \csc{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 \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>

=== [[دوال القطع الزائد]] ===
:''more integrals: [[List of integrals of hyperbolic functions]]''
:<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>

==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

:<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>

:<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^\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}}e^\frac{b^2-4ac}{4a}</math>

{{بوابة رياضيات}}

[[تصنيف:رياضيات]]
[[تصنيف:قوائم متعلقة بالرياضيات]]

[[af:Lys van integrale]]
[[bs:Tabela integrala]]
[[de:Tabelle von Ableitungs- und Stammfunktionen]]
[[fr:Table de primitives]]
[[id:Tabel integral]]
[[it:Tavola degli integrali più comuni]]
[[ko:적분표]]
[[nl:Lijst van integralen]]
[[pt:Tábua de integrais]]
[[ro:Tabel de integrale]]
[[ru:Список интегралов элементарных функций]]
[[sl:Tabela integralov]]
[[sr:Таблични интеграли]]
[[tr:İntegral tablosu]]
[[uk:Таблиця інтегралів]]

جدول التكاملات - تاريخ المراجعة

WikiSysop: ١ مراجعة: الصفحات في تصنيف رياضيات