Tabel integral

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Integrasi merupakan operasi dasar dalam kalkulus integral. Sementara diferensiasi mempunyai kaidah-kaidah mudah di mana turunan dari suatu fungsi yang rumit dapat dihitung dengan melakukan diferensiasi dari fungsi komponen yang lebih sederhana, integrasi tidak demikian, sehingga table dari integral yang sudah diketahui seringkali sangat berguna. Berikut adalah sejumlah antiderivatif yang paling umum

Artikel ini memberikan tabel operasi integrasi yang umum dijumpai. Pada daftar integrasi di bawah ini, C menyatakan konstanta sebarang.

Daftar integral[sunting | sunting sumber]

Daftar integral yang lebih detail dapat dilihat pada halaman-halaman berikut

Aturan integrasi dari fungsi-fungsi umum[sunting | sunting sumber]

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

Integral fungsi sederhana[sunting | sunting sumber]

C sering digunakan untuk arbitrary constant of integration yang hanya dapat ditentukan jika suatu nilai integral pada beberapa titik sudah diketahui. Jadi setiap fungsi mempunyai jumlah antiderivatif tidak terbatas.

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

Fungsi rasional[sunting | sunting sumber]

\int \,{\rm d}x = x + C
\int x^n\,{\rm d}x =  \frac{x^{n+1}}{n+1} + C\qquad\mbox{ jika }n \ne -1
\int {dx \over x} = \ln{\left|x\right|} + C
\int {dx \over {a^2+x^2}} = {1 \over a}\arctan {x \over a} + C

Fungsi irrasional[sunting | sunting sumber]

\int {dx \over \sqrt{a^2-x^2}} = \sin^{-1} {x \over a} + C
\int {-dx \over \sqrt{a^2-x^2}} = \cos^{-1} {x \over a} + C
\int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \sec^{-1} {|x| \over a} + C

Fungsi logaritma[sunting | sunting sumber]

\int \ln {x}\,dx = x \ln {x} - x + C
\int \log_b {x}\,dx = x \log_b {x} - x \log_b {e} + C

Fungsi eksponensial[sunting | sunting sumber]

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

Fungsi trigonometri[sunting | sunting sumber]

Artikel utama: Daftar integral dari fungsi trigonometri dan Daftar integral dari fungsi arc
\int \sin{x}\, dx = -\cos{x} + C
\int \cos{x}\, dx = \sin{x} + C
\int \tan{x} \, dx = \ln{\left| \sec {x} \right|} + C
\int \cot{x} \, dx = -\ln{\left| \csc{x} \right|} + C
\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C
\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C
\int \sec^2 x \, dx = \tan x + C
\int \csc^2 x \, dx = -\cot x + C
\int \sec{x} \, \tan{x} \, dx = \sec{x} + C
\int \csc{x} \, \cot{x} \, dx = - \csc{x} + C
\int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + C
\int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C
\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C
\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx
\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx
\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C

Fungsi hiperbolik[sunting | sunting sumber]

\int \sinh x \, dx = \cosh x + C
\int \cosh x \, dx = \sinh x + C
\int \tanh x \, dx = \ln| \cosh x | + C
\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C
\int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C
\int \coth x \, dx = \ln| \sinh x | + C

Fungsi inversi hiperbolik[sunting | sunting sumber]

\int \operatorname{arsinh} x \, dx  = x \operatorname{arsinh} x - \sqrt{x^2+1} + C
\int \operatorname{arcosh} x \, dx  = x \operatorname{arcosh} x - \sqrt{x^2-1} + C
\int \operatorname{artanh} x \, dx  = x \operatorname{artanh} x + \frac{1}{2}\log{(1-x^2)} + C
\int \operatorname{arcsch}\,x \, dx = x \operatorname{arcsch} x+ \log{\left[x\left(\sqrt{1+\frac{1}{x^2}} + 1\right)\right]} + C
\int \operatorname{arsech}\,x \, dx = x \operatorname{arsech} x- \arctan{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + C
\int \operatorname{arcoth} \, dx  = x \operatorname{arcoth} x+ \frac{1}{2}\log{(x^2-1)} + C

"Sophomore's dream"

\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}

diyakini berasal dari Johann Bernoulli.

Lihat pula[sunting | sunting sumber]

Referensi[sunting | sunting sumber]

Pustaka[sunting | sunting sumber]

  • 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[sunting | sunting sumber]

Pranala luar[sunting | sunting sumber]

Tabel integral[sunting | sunting sumber]

Derivasi[sunting | sunting sumber]

Layanan Online[sunting | sunting sumber]

Program open source[sunting | sunting sumber]