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{{about|sebagian besar integral tak tentu dalam kalkulus|daftar integral tertentu|Daftar integral tertentu}}
{{Kalkulus|Integral}}
Pengintegralan atau integrasi merupakan operasi dasar dalam [[kalkulus integral]]. Operasi lawannya, [[turunan]], mempunyai kaidah yang dapat menurunkan fungsi dengan bentuk yang lebih mudah menjadi fungsi dengan bentuk yang lebih rumit. Sayangnya, integral tidak mempunyai kaidah yang dapat menghitung sebaliknya, sehingga seringkali diperlukan tabel yang memuat kumpulan integral.
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== Perkembangan sejarah integral ==
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.
Not all [[closed-form expression]]s have closed-form antiderivatives; this study forms the subject of [[differential Galois theory]], which was initially developed by [[Joseph Liouville]] in the 1830s and 1840s, leading to [[Liouville's theorem (differential algebra)|Liouville's theorem]] which classifies which expressions have closed form antiderivatives. A simple example of a function without a closed form antiderivative is ''e''<sup>−''x''<sup>2</sup></sup>, whose antiderivative is (up to constants) the [[error function]].
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 ==
Daftar integral yang lebih detail dapat dilihat pada halaman-halaman berikut
* [[Daftar integral dari fungsi rasional]]
* [[Daftar integral dari fungsi irrasional]]
* [[Daftar integral dari fungsi trigonometri]]
* [[Daftar integral dari fungsi trigonometri terbalik]]
* [[Daftar integral dari fungsi hiperbolik]]
* [[Daftar integral dari fungsi hiperbolik terbalik]]
* [[Daftar integral dari fungsi eksponensial]]
* [[Daftar integral dari fungsi logaritmik]]
* [[Daftar integral dari fungsi Gaussian]]
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Gradshteyn, Ryzhik, Jeffrey, Zwillinger's ''Table of Integrals, Series, and Products'' contains a large collection of results. An even larger, multivolume table is the ''Integrals and Series'' by Prudnikov, Brychkov, and [[Oleg Igorevich Marichev|Marichev]] (with volumes 1–3 listing integrals and series of [[elementary function|elementary]] and [[special functions]], volume 4–5 are tables of [[Laplace transform]]s). More compact collections can be found in e.g. Brychkov, Marichev, Prudnikov's ''Tables of Indefinite Integrals'', or as chapters in Zwillinger's ''CRC Standard Mathematical Tables and Formulae'', Bronstein and Semendyayev's ''Handbook of Mathematics'' (Springer) and ''Oxford Users' Guide to Mathematics'' (Oxford Univ. Press), and other mathematical handbooks.
Other useful resources include [[Abramowitz and Stegun]] and the [[Bateman Manuscript Project]]. Both works contain many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. Two volumes of the Bateman Manuscript are specific to integral transforms.
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 ==
Baris 13 ⟶ 44:
:# <math>\int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + C </math>
== Integral
Konstanta ''C'' sering digunakan untuk [[Konstanta integrasi|konstanta sembarang]] dalam integrasi. Konstanta ini hanya dapat ditentukan jika suatu nilai integral pada beberapa titik sudah diketahui. Jadi, setiap fungsi mempunyai jumlah integral tidak terbatas.
Rumus-rumus berikut hanya menyatakan dalam bentuk lain pernyataan-pernyataan dalam [[tabel turunan]].
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=== Integrals with a singularity ===
When there is a singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), then ''C'' does not need to be the same on both sides of the singularity. The forms below normally assume the [[Cauchy principal value]] around a singularity in the value of ''C'' but this is not in general necessary. For instance in
::<math>\int {1 \over x}\,dx = \ln \left|x \right| + C</math>
there is a singularity at 0 and the antiderivative becomes infinite there. If the integral above would be used to compute a definite integral between -1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −''i''{{pi}} when using a path above the origin and ''i''{{pi}} for a path below the origin. A function on the real line could use a completely different value of ''C'' on either side of the origin as in:
: <math> \int {1 \over x}\,dx = \ln|x| + \begin{cases} A & \text{if }x>0; \\ B & \text{if }x < 0. \end{cases} </math>
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=== Fungsi rasional ===
{{
:<math>\int \,
:<math>\int x^n\,
:<math>\int (ax+b)^n\,dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C\qquad\mbox{ jika }n \ne -1</math>
:<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 ===
{{
:<math>\int {dx \over \sqrt{a^2-x^2}} = \
:<math>\int {-dx \over \sqrt{a^2-x^2}} = \
:<math>\int {dx \over
:<math>\int {-dx \over a^2+x^2} = {1 \over a} \arccot {x \over a} + C</math>
:<math>\int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \arcsec {|x| \over a} + C</math>
:<math>\int {-dx \over x \sqrt{x^2-a^2}} = {1 \over a} \arccsc {|x| \over a} + C</math>
=== Fungsi eksponensial ===
{{
:<math>\int e^x\,dx = e^x + C</math>
:<math>\int a^x\,dx = \frac{a^x}{\ln{a}} + C</math>
=== Fungsi logaritma ===
{{Main|Daftar integral dari fungsi logaritmik}}
:<math>\int \ln {x}\,dx = x \ln {x} - x + C</math>
:<math>\int \,^b\!\log {x}\,dx = x \cdot \, ^b\!\log x - x \cdot \,^b\!\log e + C</math>
=== Fungsi trigonometri ===
:''Artikel utama: [[Daftar integral dari fungsi trigonometri
:<math>\int \sin{x}\, dx = -\cos{x} + C</math>
:<math>\int \cos{x}\, dx = \sin{x} + C</math>
Baris 54 ⟶ 103:
:<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>
=== Fungsi trigonometri terbalik ===
:''Artikel utama: [[Daftar integral dari fungsi trigonometri terbalik]]''
:<math>\int \arcsin(x) \, dx = x \, arcsin(x) + {\sqrt{1 - x^2}} + C</math>
:<math>\int \arccos(x) \, dx = x \, arccos(x) - {\sqrt{1 - x^2}} + C</math>
:<math>\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C</math>
:<math>\int \arccot{x} \, dx = x \, \arccot{x} + \frac{1}{2} \ln{\left| 1 + x^2\right|} + C</math>
:<math>\int\arcsec(x)\,dx= x\arcsec(x) \, - \,
\ln\left(\left|x\right|+\sqrt{x^2-1}\right)\,+\,C=
x\arcsec(x)-\operatorname{arcosh}|x|+C</math>
:<math>\int\arccsc(x)\,dx= x\arccsc(x) \, + \,
\ln\left(\left|x\right|+\sqrt{x^2-1}\right)\,+\,C=
x\arccsc(x)+\operatorname{arcosh}|x|+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 \mbox{sech}\,x \, dx = \arctan (\sinh x) + C</math>
:<math>\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C</math>
=== Fungsi
{{Main|Daftar integral dari fungsi hiperbolik terbalik}}
: <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{arcoth} \, dx = x \operatorname{arcoth} x+ \frac{1}{2}\log{(x^2-1)} + 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>
: <math>\int \operatorname{arcsch}\,x \, dx = x \operatorname{arcsch} x+ \log{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + C</math>
Integral lain, yaitu "''[[Sophomore's dream]]''", diyakini berasal dari [[Johann Bernoulli]]. Integral tersebut di antaranya
:<math>\begin{align}
\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 (-n)^{-n} &&(= 0,78343051071213\dots)
\end{align}</math>
== Lihat pula ==
* [[Integral]]
* [[Kalkulus]]
* [[Fungsi gamma tidak komplit]]
* [[Jumlah tak terbatas]]
* [[Daftar limit]]
* [[Daftar deret matematikal]]
* [[Integrasi simbolik]]
{{Lists of integrals}}
== Referensi ==
{{reflist}}
== Pustaka ==
* [[Milton Abramowitz|M. Abramowitz]] and [[Irene Stegun|I.A. Stegun]], editors. ''[[Abramowitz and Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]''.
* 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. [http://www.mathtable.com/gr Errata.] ''(Several previous editions as well.)''
* A.P. Prudnikov (А.П. Прудников), Yu.A. Brychkov (Ю.А. Брычков), O.I. Marichev (О.И. Маричев). ''Integrals and Series''. First edition (Russian), volume 1–5, [[Nauka (publisher)|Nauka]], 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/[[CRC Press]], 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
* Yu.A. Brychkov (Ю.А. Брычков), ''Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas''. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X.
* 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.)''
=== Sejarah ===
* Meyer Hirsch, [http://books.google.com/books?id=Cdg2AAAAMAAJ Integraltafeln, oder, Sammlung von Integralformeln] (Duncker und Humblot, Berlin, 1810)
* Meyer Hirsch, [http://books.google.com/books?id=NsI2AAAAMAAJ Integral Tables, Or, A Collection of Integral Formulae] (Baynes and son, London, 1823) [English translation of ''Integraltafeln'']
* David Bierens de Haan, [http://www.archive.org/details/nouvetaintegral00haanrich Nouvelles Tables d'Intégrales définies] (Engels, Leiden, 1862)
* Benjamin O. Pierce [http://books.google.com/books?id=pYMRAAAAYAAJ A short table of integrals – revised edition] (Ginn & co., Boston, 1899)
== Pranala luar ==
=== Tabel integral ===
* [http://tutorial.math.lamar.edu/pdf/Common_Derivatives_Integrals.pdf Paul's Online Math Notes]
* A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): [http://pi.physik.uni-bonn.de/~dieckman/IntegralsIndefinite/IndefInt.html Indefinite Integrals] [http://pi.physik.uni-bonn.de/~dieckman/IntegralsDefinite/DefInt.html Definite Integrals]
* [http://mathmajor.org/calculus-and-analysis/table-of-integrals/ Math Major: A Table of Integrals] {{Webarchive|url=https://archive.today/20121030002907/http://mathmajor.org/calculus-and-analysis/table-of-integrals/ |date=2012-10-30 }}
* {{cite web | last1=O'Brien |first1=Francis J. Jr. | url=http://www.docstoc.com/docs/23969109/500-Integrals-of-Elementary-and-Special-Functions |title=500 Integrals}} Derived integrals of exponential and logarithmic functions
* [http://www.apmaths.uwo.ca/RuleBasedMathematics/index.html Rule-based Mathematics] Precisely defined indefinite integration rules covering a wide class of integrands
* {{cite arxiv| first1= Richard J. | last1=Mathar | title=Yet another table of integrals | eprint=1207.5845 |year=2012}}
=== Derivasi ===
* [http://www.math.tulane.edu/~vhm/Table.html V. H. Moll, The Integrals in Gradshteyn and Ryzhik]
=== Layanan daring ===
* [http://www.wolframalpha.com/examples/Integrals.html Integration examples for Wolfram Alpha]
=== Program open source ===
* [http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page wxmaxima gui for Symbolic and numeric resolution of many mathematical problems] {{Webarchive|url=https://web.archive.org/web/20110320113320/http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page |date=2011-03-20 }}
[[Kategori:Daftar matematika|Integral]]
[[Kategori:Tabel matematika|Integral]]
[[Kategori:Kalkulus]]
[[Kategori:Integral]]
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