Tabel integral 📖 Wikipedia

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, ${\displaystyle C}$ mengartikan konstanta sembarang.

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

1. ${\displaystyle \int af(x)\,dx=a\int f(x)\,dx\qquad {\mbox{(}}a{\mbox{ konstan)}}\,\!}$
2. ${\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}$
3. ${\displaystyle \int f(x)g(x)\,dx=f(x)\int g(x)\,dx-\int \left[f'(x)\left(\int g(x)\,dx\right)\right]\,dx}$
4. ${\displaystyle \int [f(x)]^{n}f'(x)\,dx={[f(x)]^{n+1} \over n+1}+C\qquad {\mbox{(untuk }}n\neq -1{\mbox{)}}\,\!}$
5. ${\displaystyle \int {f'(x) \over f(x)}\,dx=\ln {\left|f(x)\right|}+C}$
6. ${\displaystyle \int {f'(x)f(x)}\,dx={1 \over 2}[f(x)]^{2}+C}$

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.

${\displaystyle \int \,dx=x+C}$

${\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ jika }}n\neq -1}$

${\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\mbox{ jika }}n\neq -1}$

${\displaystyle \int {dx \over x}=\ln {\left|x\right|}+C}$

${\displaystyle \int {dx \over {a^{2}+x^{2}}}={1 \over a}\arctan {x \over a}+C}$

${\displaystyle \int {dx \over {\sqrt {a^{2}-x^{2}}}}=\arcsin {x \over a}+C}$

${\displaystyle \int {-dx \over {\sqrt {a^{2}-x^{2}}}}=\arccos {x \over a}+C}$

${\displaystyle \int {dx \over a^{2}+x^{2}}={1 \over a}\arctan {x \over a}+C}$

${\displaystyle \int {-dx \over a^{2}+x^{2}}={1 \over a}\operatorname {arccot} {x \over a}+C}$

${\displaystyle \int {dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\operatorname {arcsec} {|x| \over a}+C}$

${\displaystyle \int {-dx \over x{\sqrt {x^{2}-a^{2}}}}={1 \over a}\operatorname {arccsc} {|x| \over a}+C}$

${\displaystyle \int e^{x}\,dx=e^{x}+C}$

${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C}$

${\displaystyle \int \ln {x}\,dx=x\ln {x}-x+C}$

${\displaystyle \int \,^{b}\!\log {x}\,dx=x\cdot \,^{b}\!\log x-x\cdot \,^{b}\!\log e+C}$

Artikel utama: Daftar integral dari fungsi trigonometri

${\displaystyle \int \sin {x}\,dx=-\cos {x}+C}$

${\displaystyle \int \cos {x}\,dx=\sin {x}+C}$

${\displaystyle \int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C}$

${\displaystyle \int \cot {x}\,dx=-\ln {\left|\csc {x}\right|}+C}$

${\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}$

${\displaystyle \int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C}$

${\displaystyle \int \sec ^{2}x\,dx=\tan x+C}$

${\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}$

${\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}$

${\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}$

${\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}(x-\sin x\cos x)+C}$

${\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}(x+\sin x\cos x)+C}$

${\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}$

${\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}$

${\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}$

Artikel utama: Daftar integral dari fungsi trigonometri terbalik

${\displaystyle \int \arcsin(x)\,dx=x\,arcsin(x)+{\sqrt {1-x^{2}}}+C}$

${\displaystyle \int \arccos(x)\,dx=x\,arccos(x)-{\sqrt {1-x^{2}}}+C}$

${\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}$

${\displaystyle \int \operatorname {arccot} {x}\,dx=x\,\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}$

${\displaystyle \int \operatorname {arcsec}(x)\,dx=x\operatorname {arcsec}(x)\,-\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arcsec}(x)-\operatorname {arcosh} |x|+C}$

${\displaystyle \int \operatorname {arccsc}(x)\,dx=x\operatorname {arccsc}(x)\,+\,\ln \left(\left|x\right|+{\sqrt {x^{2}-1}}\right)\,+\,C=x\operatorname {arccsc}(x)+\operatorname {arcosh} |x|+C}$

${\displaystyle \int \sinh x\,dx=\cosh x+C}$

${\displaystyle \int \cosh x\,dx=\sinh x+C}$

${\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}$

${\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}$

${\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C}$

${\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}$

${\displaystyle \int \operatorname {arsinh} x\,dx=x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}+C}$

${\displaystyle \int \operatorname {arcosh} x\,dx=x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}+C}$

${\displaystyle \int \operatorname {artanh} x\,dx=x\operatorname {artanh} x+{\frac {1}{2}}\log {(1-x^{2})}+C}$

${\displaystyle \int \operatorname {arcoth} \,dx=x\operatorname {arcoth} x+{\frac {1}{2}}\log {(x^{2}-1)}+C}$

${\displaystyle \int \operatorname {arsech} \,x\,dx=x\operatorname {arsech} x-\arctan {\left({\frac {x}{x-1}}{\sqrt {\frac {1-x}{1+x}}}\right)}+C}$

${\displaystyle \int \operatorname {arcsch} \,x\,dx=x\operatorname {arcsch} x+\log {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C}$

Integral lain, yaitu "Sophomore's dream", diyakini berasal dari Johann Bernoulli. Integral tersebut di antaranya

${\displaystyle {\begin{aligned}\int _{0}^{1}x^{-x}\,dx&=\sum _{n=1}^{\infty }n^{-n}&&(=1,29128599706266\dots )\\\int _{0}^{1}x^{x}\,dx&=-\sum _{n=1}^{\infty }(-n)^{-n}&&(=0,78343051071213\dots )\end{aligned}}}$

• Integral
• Kalkulus
• Fungsi gamma tidak komplit
• Jumlah tak terbatas
• Daftar limit
• Daftar deret matematikal
• Integrasi simbolik

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• 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.)

• 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)

• 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. 

• V. H. Moll, The Integrals in Gradshteyn and Ryzhik

• Integration examples for Wolfram Alpha

• wxmaxima gui for Symbolic and numeric resolution of many mathematical problems Diarsipkan 2011-03-20 di Wayback Machine.