Rabu, 08 Agustus 2012

Integral

Aturan integrasi dari fungsi-fungsi umum

  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 dari fungsi-fungsi sederhana

Fungsi rasional

\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

!Artikel utama untuk bagian ini adalah: Daftar integral dari fungsi irrasional
\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

Logaritma


\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

!Artikel utama untuk bagian ini adalah: Daftar integral dari fungsi eksponensial
\int e^x\,dx = e^x + C
\int a^x\,dx = \frac{a^x}{\ln{a}} + C

Fungsi trigonometri

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

!Artikel utama untuk bagian ini adalah: Daftar integral dari fungsi hiperbolik
\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

\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
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