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{{about|sebagian besar integral tak tentu dalam kalkulus|daftar integral tertentu|Daftar integral tertentu}}
{{Kalkulus|Integral}}
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Baris 13:
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
Baris 44 ⟶ 45:
== Integral fungsi sederhana ==
Konstanta ''C'' sering digunakan untuk [[
Rumus-rumus berikut hanya menyatakan dalam bentuk lain pernyataan-pernyataan dalam [[tabel turunan]].
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=== Integrals with a singularity ===
Baris 56 ⟶ 58:
: <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 ===
{{Main|Daftar integral dari 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>
Baris 65 ⟶ 69:
=== Fungsi irrasional ===
{{Main|Daftar integral dari fungsi irrasional}}
:<math>\int {dx \over \sqrt{a^2-x^2}} = \
:<math>\int {-dx \over \sqrt{a^2-x^2}} = \
:<math>\int {dx \over a^2+x^2} = {1 \over a} \
:<math>\int {-dx \over a^2+x^2} = {1 \over a} \
:<math>\int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \
:<math>\int {-dx \over x \sqrt{x^2-a^2}} = {1 \over a} \
=== Fungsi logaritma ===▼
{{Main|Daftar integral dari fungsi logaritmik}}▼
:<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>▼
=== Fungsi eksponensial ===
Baris 81 ⟶ 80:
:<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 99 ⟶ 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 \
:<math>\int \arctan{x} \, dx = x \, \arctan{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 ===
Baris 106 ⟶ 123:
:<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>
:<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 ===
Baris 115 ⟶ 132:
: <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{
: <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>
▲: <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{arcoth} \, dx = x \operatorname{arcoth} x+ \frac{1}{2}\log{(x^2-1)} + 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
\int_0^1 x^x \,dx &= -\sum_{n=1}^\infty (-n)^{-n} &&(= 0
\end{align}</math>
== Lihat pula ==
Baris 133 ⟶ 148:
* [[Jumlah tak terbatas]]
* [[Daftar limit]]
* [[Daftar deret
* [[Integrasi simbolik]]
{{Lists of integrals}}
Baris 157 ⟶ 172:
* [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.
* {{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
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