Tabel integral - Riwayat revisi

Maintenance script: Presentation V4: sitasi, referensi, Math, Wikimedia Commons, dan atribusi

2026-08-24T23:08:54Z

Presentation V4: sitasi, referensi, Math, Wikimedia Commons, dan atribusi

← Revisi sebelumnya		Revisi per 24 Agustus 2026 23.08
Baris 2:		Baris 2:

	Berikut adalah daftar yang memuat integral atau antiturunan yang paling umum dijumpai. Pada daftar di bawah ini, <math>C</math> mengartikan konstanta sembarang.		Berikut adalah daftar yang memuat integral atau antiturunan yang paling umum dijumpai. Pada daftar di bawah ini, <math>C</math> mengartikan konstanta sembarang.


	== Daftar integral ==		== Daftar integral ==
Baris 16:		Baris 15:
	* [[Daftar integral dari fungsi logaritmik]]		* [[Daftar integral dari fungsi logaritmik]]
	* [[Daftar integral dari fungsi Gaussian]]		* [[Daftar integral dari fungsi Gaussian]]


	== Aturan integrasi dari fungsi-fungsi umum ==		== Aturan integrasi dari fungsi-fungsi umum ==

	:# <math>\int af(x)\,dx = a\int f(x)\,dx \qquad\mbox{(}a \mbox{ konstan)}\,\!</math>		:# <math>\int af(x)\,dx = a\int f(x)\,dx \qquad\mbox{(}a \mbox{ konstan)}\,\!</math>
	:# <math>\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx</math>		:# <math>\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx</math>
Baris 31:		Baris 28:

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


	=== Fungsi rasional ===		=== Fungsi rasional ===

	:<math>\int \, dx = x + C</math>		:<math>\int \, dx = x + C</math>
	:<math>\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ jika }n \ne -1</math>		:<math>\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ jika }n \ne -1</math>
Baris 42:		Baris 37:

	=== Fungsi irrasional ===		=== Fungsi irrasional ===

	:<math>\int {dx \over \sqrt{a^2-x^2}} = \arcsin {x \over a} + C</math>		:<math>\int {dx \over \sqrt{a^2-x^2}} = \arcsin {x \over a} + C</math>
	:<math>\int {-dx \over \sqrt{a^2-x^2}} = \arccos {x \over a} + C</math>		:<math>\int {-dx \over \sqrt{a^2-x^2}} = \arccos {x \over a} + C</math>
	:<math>\int {dx \over a^2+x^2} = {1 \over a} \arctan {x \over a} + C</math>		:<math>\int {dx \over a^2+x^2} = {1 \over a} \arctan {x \over a} + C</math>
	:<math>\int {-dx \over a^2+x^2} = {1 \over a} \arccot {x \over a} + C</math>		:<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 + 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 ===
			:<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>
			:<math>\int \tan{x} \, dx = \ln{\left\| \sec {x} \right\|} + C</math>
	:<math>\int \cot{x} \, dx = -\ln{\left\| \csc{x} \right\|} + C</math>		:<math>\int \cot{x} \, dx = -\ln{\left\| \csc{x} \right\|} + C</math>
	:<math>\int \sec{x} \, dx = \ln{\left\| \sec{x} + \tan{x}\right\|} + C</math>		:<math>\int \sec{x} \, dx = \ln{\left\| \sec{x} + \tan{x}\right\|} + C</math>
Baris 76:		Baris 85:

	=== Fungsi hiperbolik ===		=== Fungsi hiperbolik ===

	:<math>\int \sinh x \, dx = \cosh x + C</math>		:<math>\int \sinh x \, dx = \cosh x + C</math>
	:<math>\int \cosh x \, dx = \sinh x + C</math>		:<math>\int \cosh x \, dx = \sinh x + C</math>
Baris 85:		Baris 93:

	=== Fungsi hiperbolik terbalik ===		=== Fungsi hiperbolik terbalik ===

	: <math>\int \operatorname{arsinh} x \, dx = x \operatorname{arsinh} x - \sqrt{x^2+1} + 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{arcosh} x \, dx = x \operatorname{arcosh} x - \sqrt{x^2-1} + C</math>
Baris 107:		Baris 114:
	* [[Daftar deret matematikal]]		* [[Daftar deret matematikal]]
	* [[Integrasi simbolik]]		* [[Integrasi simbolik]]


	~~== Referensi ==~~


	== Pustaka ==		== Pustaka ==
Baris 129:		Baris 132:
	* [http://tutorial.math.lamar.edu/pdf/Common_Derivatives_Integrals.pdf Paul's Online Math Notes]		* [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]		* 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]		* [http://mathmajor.org/calculus-and-analysis/table-of-integrals/ Math Major: A Table of Integrals]
	* Derived integrals of exponential and logarithmic functions		* 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		* [http://www.apmaths.uwo.ca/RuleBasedMathematics/index.html Rule-based Mathematics] Precisely defined indefinite integration rules covering a wide class of integrands
	*		*

	=== Derivasi ===		=== Derivasi ===
Baris 142:		Baris 145:
	=== Program open source ===		=== Program open source ===
	* [http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page wxmaxima gui for Symbolic and numeric resolution of many mathematical problems]		* [http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page wxmaxima gui for Symbolic and numeric resolution of many mathematical problems]



	== Sumber dan atribusi ==		== Sumber dan atribusi ==

	Konten artikel ini diadaptasi dari [https://id.wikipedia.org/w/index.php?title=Tabel+integral&oldid=28593848 Wikipedia bahasa Indonesia], revisi 28593848 (2025-11-21T20:55:27Z), yang tersedia berdasarkan lisensi Creative Commons Atribusi-BerbagiSerupa (CC BY-SA). Mohon gunakan konten ini secara bijak serta sesuai dengan ketentuan lisensi yang berlaku.		Konten artikel ini diadaptasi dari [https://id.wikipedia.org/w/index.php?title=Tabel+integral&oldid=28593848 Wikipedia bahasa Indonesia], revisi 28593848 (2025-11-21T20:55:27Z), yang tersedia berdasarkan lisensi Creative Commons Atribusi-BerbagiSerupa (CC BY-SA). Mohon gunakan konten ini secara bijak serta sesuai dengan ketentuan lisensi yang berlaku.

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Maintenance script: Impor teks terkontrol dari Wikipedia bahasa Indonesia; revisi 28593848; atribusi sumber disertakan.

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Impor teks terkontrol dari Wikipedia bahasa Indonesia; revisi 28593848; atribusi sumber disertakan.

Halaman baru

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 sering kali diperlukan tabel yang memuat kumpulan integral.

Berikut adalah daftar yang memuat integral atau antiturunan yang paling umum dijumpai. Pada daftar di bawah ini, <math>C</math> mengartikan konstanta sembarang.

== 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]]

== Aturan integrasi dari fungsi-fungsi umum ==

:# <math>\int af(x)\,dx = a\int f(x)\,dx \qquad\mbox{(}a \mbox{ konstan)}\,\!</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[f'(x) \left(\int g(x)\,dx\right)\right]\,dx</math>
:# <math>\int [f(x)]^n f'(x)\,dx = {[f(x)]^{n+1} \over n+1} + C \qquad\mbox{(untuk } n\neq -1\mbox{)}\,\! </math>
:# <math>\int {f'(x)\over f(x)}\,dx= \ln{\left|f(x)\right|} + C </math>
:# <math>\int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + C </math>

== Integral fungsi sederhana ==
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]].

=== Fungsi rasional ===

:<math>\int \, dx = x + C</math>
:<math>\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ jika }n \ne -1</math>
:<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}} = \arcsin {x \over a} + C</math>
:<math>\int {-dx \over \sqrt{a^2-x^2}} = \arccos {x \over a} + C</math>
:<math>\int {dx \over a^2+x^2} = {1 \over a} \arctan {x \over a} + C</math>
:<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 + C</math>
:<math>\int \cot{x} \, dx = -\ln{\left| \csc{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 \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan 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>

=== 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 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]]

== Referensi ==

== 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]
* 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
*

=== 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]

== Sumber dan atribusi ==

Konten artikel ini diadaptasi dari [https://id.wikipedia.org/w/index.php?title=Tabel+integral&oldid=28593848 Wikipedia bahasa Indonesia], revisi 28593848 (2025-11-21T20:55:27Z), yang tersedia berdasarkan lisensi Creative Commons Atribusi-BerbagiSerupa (CC BY-SA). Mohon gunakan konten ini secara bijak serta sesuai dengan ketentuan lisensi yang berlaku.