- Source: Tabel 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.
Berikut adalah daftar yang memuat integral atau antiturunan yang paling umum dijumpai. Pada daftar di bawah ini,
C
{\displaystyle C}
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
∫
a
f
(
x
)
d
x
=
a
∫
f
(
x
)
d
x
(
a
konstan)
{\displaystyle \int af(x)\,dx=a\int f(x)\,dx\qquad {\mbox{(}}a{\mbox{ konstan)}}\,\!}
∫
[
f
(
x
)
+
g
(
x
)
]
d
x
=
∫
f
(
x
)
d
x
+
∫
g
(
x
)
d
x
{\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}
∫
f
(
x
)
g
(
x
)
d
x
=
f
(
x
)
∫
g
(
x
)
d
x
−
∫
[
f
′
(
x
)
(
∫
g
(
x
)
d
x
)
]
d
x
{\displaystyle \int f(x)g(x)\,dx=f(x)\int g(x)\,dx-\int \left[f'(x)\left(\int g(x)\,dx\right)\right]\,dx}
∫
[
f
(
x
)
]
n
f
′
(
x
)
d
x
=
[
f
(
x
)
]
n
+
1
n
+
1
+
C
(untuk
n
≠
−
1
)
{\displaystyle \int [f(x)]^{n}f'(x)\,dx={[f(x)]^{n+1} \over n+1}+C\qquad {\mbox{(untuk }}n\neq -1{\mbox{)}}\,\!}
∫
f
′
(
x
)
f
(
x
)
d
x
=
ln
|
f
(
x
)
|
+
C
{\displaystyle \int {f'(x) \over f(x)}\,dx=\ln {\left|f(x)\right|}+C}
∫
f
′
(
x
)
f
(
x
)
d
x
=
1
2
[
f
(
x
)
]
2
+
C
{\displaystyle \int {f'(x)f(x)}\,dx={1 \over 2}[f(x)]^{2}+C}
Integral fungsi sederhana
Konstanta C sering digunakan untuk 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
=∫
d
x
=
x
+
C
{\displaystyle \int \,dx=x+C}
∫
x
n
d
x
=
x
n
+
1
n
+
1
+
C
jika
n
≠
−
1
{\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ jika }}n\neq -1}
∫
(
a
x
+
b
)
n
d
x
=
(
a
x
+
b
)
n
+
1
a
(
n
+
1
)
+
C
jika
n
≠
−
1
{\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\mbox{ jika }}n\neq -1}
∫
d
x
x
=
ln
|
x
|
+
C
{\displaystyle \int {dx \over x}=\ln {\left|x\right|}+C}
∫
d
x
a
2
+
x
2
=
1
a
arctan
x
a
+
C
{\displaystyle \int {dx \over {a^{2}+x^{2}}}={1 \over a}\arctan {x \over a}+C}
= Fungsi irrasional
=∫
d
x
a
2
−
x
2
=
arcsin
x
a
+
C
{\displaystyle \int {dx \over {\sqrt {a^{2}-x^{2}}}}=\arcsin {x \over a}+C}
∫
−
d
x
a
2
−
x
2
=
arccos
x
a
+
C
{\displaystyle \int {-dx \over {\sqrt {a^{2}-x^{2}}}}=\arccos {x \over a}+C}
∫
d
x
a
2
+
x
2
=
1
a
arctan
x
a
+
C
{\displaystyle \int {dx \over a^{2}+x^{2}}={1 \over a}\arctan {x \over a}+C}
∫
−
d
x
a
2
+
x
2
=
1
a
arccot
x
a
+
C
{\displaystyle \int {-dx \over a^{2}+x^{2}}={1 \over a}\operatorname {arccot} {x \over a}+C}
∫
d
x
x
x
2
−
a
2
=
1
a
arcsec
|
x
|
a
+
C
{\displaystyle \int {dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\operatorname {arcsec} {|x| \over a}+C}
∫
−
d
x
x
x
2
−
a
2
=
1
a
arccsc
|
x
|
a
+
C
{\displaystyle \int {-dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\operatorname {arccsc} {|x| \over a}+C}
= Fungsi eksponensial
=∫
e
x
d
x
=
e
x
+
C
{\displaystyle \int e^{x}\,dx=e^{x}+C}
∫
a
x
d
x
=
a
x
ln
a
+
C
{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C}
= Fungsi logaritma
=∫
ln
x
d
x
=
x
ln
x
−
x
+
C
{\displaystyle \int \ln {x}\,dx=x\ln {x}-x+C}
∫
b
log
x
d
x
=
x
⋅
b
log
x
−
x
⋅
b
log
e
+
C
{\displaystyle \int \,^{b}\!\log {x}\,dx=x\cdot \,^{b}\!\log x-x\cdot \,^{b}\!\log e+C}
= Fungsi trigonometri
=Artikel utama: Daftar integral dari fungsi trigonometri
∫
sin
x
d
x
=
−
cos
x
+
C
{\displaystyle \int \sin {x}\,dx=-\cos {x}+C}
∫
cos
x
d
x
=
sin
x
+
C
{\displaystyle \int \cos {x}\,dx=\sin {x}+C}
∫
tan
x
d
x
=
ln
|
sec
x
|
+
C
{\displaystyle \int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C}
∫
cot
x
d
x
=
−
ln
|
csc
x
|
+
C
{\displaystyle \int \cot {x}\,dx=-\ln {\left|\csc {x}\right|}+C}
∫
sec
x
d
x
=
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}
∫
csc
x
d
x
=
−
ln
|
csc
x
+
cot
x
|
+
C
{\displaystyle \int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C}
∫
sec
2
x
d
x
=
tan
x
+
C
{\displaystyle \int \sec ^{2}x\,dx=\tan x+C}
∫
csc
2
x
d
x
=
−
cot
x
+
C
{\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}
∫
sec
x
tan
x
d
x
=
sec
x
+
C
{\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}
∫
csc
x
cot
x
d
x
=
−
csc
x
+
C
{\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}
∫
sin
2
x
d
x
=
1
2
(
x
−
sin
x
cos
x
)
+
C
{\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}(x-\sin x\cos x)+C}
∫
cos
2
x
d
x
=
1
2
(
x
+
sin
x
cos
x
)
+
C
{\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}(x+\sin x\cos x)+C}
∫
sec
3
x
d
x
=
1
2
sec
x
tan
x
+
1
2
ln
|
sec
x
+
tan
x
|
+
C
{\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}
∫
sin
n
x
d
x
=
−
sin
n
−
1
x
cos
x
n
+
n
−
1
n
∫
sin
n
−
2
x
d
x
{\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}
∫
cos
n
x
d
x
=
cos
n
−
1
x
sin
x
n
+
n
−
1
n
∫
cos
n
−
2
x
d
x
{\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}
= Fungsi trigonometri terbalik
=Artikel utama: Daftar integral dari fungsi trigonometri terbalik
∫
arcsin
(
x
)
d
x
=
x
a
r
c
s
i
n
(
x
)
+
1
−
x
2
+
C
{\displaystyle \int \arcsin(x)\,dx=x\,arcsin(x)+{\sqrt {1-x^{2}}}+C}
∫
arccos
(
x
)
d
x
=
x
a
r
c
c
o
s
(
x
)
−
1
−
x
2
+
C
{\displaystyle \int \arccos(x)\,dx=x\,arccos(x)-{\sqrt {1-x^{2}}}+C}
∫
arctan
x
d
x
=
x
arctan
x
−
1
2
ln
|
1
+
x
2
|
+
C
{\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}
∫
arccot
x
d
x
=
x
arccot
x
+
1
2
ln
|
1
+
x
2
|
+
C
{\displaystyle \int \operatorname {arccot} {x}\,dx=x\,\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}
∫
arcsec
(
x
)
d
x
=
x
arcsec
(
x
)
−
ln
(
|
x
|
+
x
2
−
1
)
+
C
=
x
arcsec
(
x
)
−
arcosh
|
x
|
+
C
{\displaystyle \int \operatorname {arcsec}(x)\,dx=x\operatorname {arcsec}(x)\,-\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arcsec}(x)-\operatorname {arcosh} |x|+C}
∫
arccsc
(
x
)
d
x
=
x
arccsc
(
x
)
+
ln
(
|
x
|
+
x
2
−
1
)
+
C
=
x
arccsc
(
x
)
+
arcosh
|
x
|
+
C
{\displaystyle \int \operatorname {arccsc}(x)\,dx=x\operatorname {arccsc}(x)\,+\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arccsc}(x)+\operatorname {arcosh} |x|+C}
= Fungsi hiperbolik
=∫
sinh
x
d
x
=
cosh
x
+
C
{\displaystyle \int \sinh x\,dx=\cosh x+C}
∫
cosh
x
d
x
=
sinh
x
+
C
{\displaystyle \int \cosh x\,dx=\sinh x+C}
∫
tanh
x
d
x
=
ln
|
cosh
x
|
+
C
{\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}
∫
coth
x
d
x
=
ln
|
sinh
x
|
+
C
{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}
∫
sech
x
d
x
=
arctan
(
sinh
x
)
+
C
{\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C}
∫
csch
x
d
x
=
ln
|
tanh
x
2
|
+
C
{\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}
= Fungsi hiperbolik terbalik
=∫
arsinh
x
d
x
=
x
arsinh
x
−
x
2
+
1
+
C
{\displaystyle \int \operatorname {arsinh} x\,dx=x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}+C}
∫
arcosh
x
d
x
=
x
arcosh
x
−
x
2
−
1
+
C
{\displaystyle \int \operatorname {arcosh} x\,dx=x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}+C}
∫
artanh
x
d
x
=
x
artanh
x
+
1
2
log
(
1
−
x
2
)
+
C
{\displaystyle \int \operatorname {artanh} x\,dx=x\operatorname {artanh} x+{\frac {1}{2}}\log {(1-x^{2})}+C}
∫
arcoth
d
x
=
x
arcoth
x
+
1
2
log
(
x
2
−
1
)
+
C
{\displaystyle \int \operatorname {arcoth} \,dx=x\operatorname {arcoth} x+{\frac {1}{2}}\log {(x^{2}-1)}+C}
∫
arsech
x
d
x
=
x
arsech
x
−
arctan
(
x
x
−
1
1
−
x
1
+
x
)
+
C
{\displaystyle \int \operatorname {arsech} \,x\,dx=x\operatorname {arsech} x-\arctan {\left({\frac {x}{x-1}}{\sqrt {\frac {1-x}{1+x}}}\right)}+C}
∫
arcsch
x
d
x
=
x
arcsch
x
+
log
[
x
(
1
+
1
x
2
+
1
)
]
+
C
{\displaystyle \int \operatorname {arcsch} \,x\,dx=x\operatorname {arcsch} x+\log {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C}
Integral lain, yaitu "Sophomore's dream", diyakini berasal dari Johann Bernoulli. Integral tersebut di antaranya
∫
0
1
x
−
x
d
x
=
∑
n
=
1
∞
n
−
n
(
=
1
,
29128599706266
…
)
∫
0
1
x
x
d
x
=
−
∑
n
=
1
∞
(
−
n
)
−
n
(
=
0
,
78343051071213
…
)
{\displaystyle {\begin{aligned}\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{aligned}}}
Lihat pula
Integral
Kalkulus
Fungsi gamma tidak komplit
Jumlah tak terbatas
Daftar limit
Daftar deret matematikal
Integrasi simbolik
Referensi
Pustaka
M. Abramowitz and I.A. Stegun, editors. 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. 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, 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, Integraltafeln, oder, Sammlung von Integralformeln (Duncker und Humblot, Berlin, 1810)
Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
David Bierens de Haan, Nouvelles Tables d'Intégrales définies (Engels, Leiden, 1862)
Benjamin O. Pierce A short table of integrals – revised edition (Ginn & co., Boston, 1899)
Pranala luar
= Tabel integral
=Paul's Online Math Notes
A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): Indefinite Integrals Definite Integrals
Math Major: A Table of Integrals Diarsipkan 2012-10-30 di Archive.is
O'Brien, Francis J. Jr. "500 Integrals". Derived integrals of exponential and logarithmic functions
Rule-based Mathematics Precisely defined indefinite integration rules covering a wide class of integrands
Mathar, Richard J. (2012). "Yet another table of integrals". arΧiv:1207.5845.
= Derivasi
=V. H. Moll, The Integrals in Gradshteyn and Ryzhik
= Layanan daring
=Integration examples for Wolfram Alpha
= Program open source
=wxmaxima gui for Symbolic and numeric resolution of many mathematical problems Diarsipkan 2011-03-20 di Wayback Machine.
Kata Kunci Pencarian:
- Integral
- Tabel integral
- Integral substitusi
- Integral lipat
- Integral tak tentu
- Tabel turunan
- Daftar integral dari fungsi invers hiperbolik
- Daftar integral dari fungsi invers trigonometri
- Daftar integral dari fungsi logaritmik
- Integral takwajar
- Cell–cell recognition
- Estonia
- Capitalism
- Carl Friedrich Gauss
- Tourism in Indonesia
- Nicolae Iorga
