Artikel ini berisi tentang sebagian besar integral tak tentu dalam kalkulus. Untuk daftar integral tertentu, lihat
Daftar integral tertentu .
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,
C
{\displaystyle C}
Daftar integral
Daftar integral yang lebih detail dapat dilihat pada halaman-halaman berikut
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 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
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
Derivasi
Layanan daring
Program open source
![{\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc2c6e2acadf432d22ca42bc6a21af25e48e64d)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054fd882152acf3191c73c0a0eda7256b2ecac74)
![{\displaystyle \int [f(x)]^{n}f'(x)\,dx={[f(x)]^{n+1} \over n+1}+C\qquad {\mbox{(untuk }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fee3d8370de7075ba5cd461320474837c6282f46)
![{\displaystyle \int {f'(x)f(x)}\,dx={1 \over 2}[f(x)]^{2}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ddd18a718860218bb3d10a0ccc72af8e420c58)
![{\displaystyle \int \operatorname {arcsch} \,x\,dx=x\operatorname {arcsch} x+\log {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fd585584498c57abf914e0c56813a43dbacf8c)
