Tabel integral: Perbedaan antara revisi

 

 

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A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician [[Meyer Hirsch]] in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician [[David de Bierens de Haan]]. A new edition was published in 1862. These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by de Bierens are denoted by BI.

 

 

Since 1968 there is the [[Risch algorithm]] for determining indefinite integrals that can be expressed in term of [[elementary function]]s, typically using a [[computer algebra system]]. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the [[Meijer G-function]].

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Daftar integral yang lebih detail dapat dilihat pada halaman-halaman berikut

 

* [[Daftar integral dari fungsi irrasional]]

* [[Daftar integral dari fungsi trigonometri]]

* [[Daftar integral dari fungsi logaritmik]]

* [[Daftar integral dari fungsi Gaussian]]

There are several web sites which have tables of integrals and integrals on demand. [[Wolfram Alpha]] can show results, and for some simpler expressions, also the intermediate steps of the integration. [[Wolfram Research]] also operates another online service, the [http://integrals.wolfram.com/index.jsp Wolfram Mathematica Online Integrator].

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== Aturan integrasi dari fungsi-fungsi umum ==

 

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

 

 

Rumus-rumus berikut hanya menyatakan dalam bentuk lain pernyataan-pernyataan dalam [[tabel turunan]].

 

=== Fungsi rasional ===

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

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

 

=== Fungsi irrasional ===

 

=== Fungsi eksponensial ===

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

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

 

=== Fungsi trigonometri ===

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

:<math>\int \cos{x}\, dx = \sin{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>

 

=== Fungsi hiperbolik ===

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

 

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

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

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

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

 

== Lihat pula ==

* [[Integral]]

* [[Kalkulus]]

{{Lists of integrals}}