welcome too hell: Tabel integral

Integrasi adalah salah satu dari dua operasi dasar kalkulus; operasi yang lain adalah penurunan (derivasi). Pada penurunan, terdapat aturan yang menjadikan turunan dari fungsi-fungsi kompleks dapat ditelusuri dari penurunan fungsi-fungsi komponennya yang lebih sederhana. Hal ini tidak terdapat dalam integrasi sehingga tabel integral biasanya amat berguna.
Artikel ini memberikan tabel operasi integrasi yang umum dijumpai. Pada daftar integrasi di bawah ini, C menyatakan konstanta sebarang.

Aturan integrasi dari fungsi-fungsi umum

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

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

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

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

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

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

$\int e^x\,dx = e^x + C$

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

Fungsi trigonometri

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fungsi hiperbolik

$\int \sinh x \, dx = \cosh x + C$

$\int \cosh x \, dx = \sinh x + C$

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

$\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C$

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

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

Fungsi inversi hiperbolik

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

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

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

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

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

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

welcome too hell

Rabu, 14 Desember 2011

Tabel integral