aditya wisnu wardhana: Agustus 2012

RUMUS INTEGRAL

Integrasi parsial

Integral parsial menggunakan rumus sebagai berikut:

$Description: \int f'(x)g(x)\,dx = f(x)g(x) - \int f(x)g'(x)\,dx$

Contoh soal:

Cari nilai dari: $Description: \int \ln x \,dx\,$

$Description: f'(x) = 1, f(x) = x, g(x) = ln x, g'(x) = \frac{1}{x}\,$

Gunakan rumus di atas

$Description: \int \ln x\ dx = x ln x - \int x\frac{1}{x}\,dx\,$

$Description: = x ln x - \int 1\,dx\,$

$Description: = x ln x - x + C\,$

Substitusi trigonometri

Bentuk	Gunakan $Description: x = \frac{a}{b}\sin \alpha\,$
$Description: \sqrt{a^2-b^2x^2}\,$
$Description: \sqrt{a^2+b^2x^2}\,$	$Description: \!\, x = \frac{a}{b}\tan \alpha\,$
$Description: \sqrt{b^2x^2-a^2}\,$	$Description: \, x = \frac{a}{b}\sec \alpha\,$

Contoh soal:

Cari nilai dari: $Description: \int \frac{dx}{x^2\sqrt{x^2+4}}\,$

$Description: x = 2 \tan A, dx = 2 \sec^2 A\,dA\,$

$Description: \int \frac{dx}{x^2\sqrt{x^2+4}}\,$

$Description: = \int \frac {2 sec^2 A\,dA}{(2 tan A)^2\sqrt{4 + (2 tan A)^2}}\,$

$Description: = \int \frac {2 sec^2 A\,dA}{4 tan^2A\sqrt{4 + 4 tan^2A}}\,$

$Description: = \int \frac {2 sec^2 A\,dA}{4 tan^2A\sqrt{4(1+tan^2A)}}\,$

$Description: = \int \frac {2 sec^2 A\,dA}{4 tan^2A\sqrt{4 sec^2A}}\,$

$Description: = \int \frac {2 sec^2 A\,dA}{4 tan^2A.2sec A}\,$

$Description: = \int \frac {sec A\,dA}{4 tan^2A}\,$

$Description: = \frac {1}{4}\int \frac {secA\,dA}{tan^2A}\,$

$Description: = \frac {1}{4}\int \frac{cos A}{sin^2A}\,dA\,$

Cari nilai dari: $Description: \int \frac{cos A}{sin^2A}\,dA\,$ dengan menggunakan substitusi

$Description: t = sin A, dt = cos A\,dA\,$

$Description: \int \frac{cos A}{sin^2A}\,dA\,$

$Description: = \int \frac{dt}{t^2}\,$

$Description: = \int t^{-2}\,dt\,$

$Description: = -t^{-1} + C= -\frac{1}{sin A} + C\,$

Masukkan nilai tersebut:

$Description: = \frac {1}{4}\int \frac{cos A}{sin^2A}\,dA\,$

$Description: = \frac {1}{4}.-\frac{1}{sin A} + C\,$

$Description: = -\frac {1}{4 sin A} + C\,$

Nilai sin A adalah $Description: \frac{x}{\sqrt{x^2+4}}$

$Description: = -\frac {1}{4 sin A} + C\,$

$Description: = -\frac {\sqrt{x^2+4}}{4x} + C\,$

Integrasi pecahan parsial

Contoh soal:

Cari nilai dari: $Description: \int\frac{dx}{x^2-4}\,$

$Description: \frac{1}{x^2-4} = \frac{A}{x+2} + \frac{B}{x-2}\,$

Akan diperoleh dua persamaan yaitu dan $Description: A-B = -\frac{1}{2}$

Dengan menyelesaikan kedua persamaan akan diperoleh hasil $Description: A = -\frac{1}{4}, B = \frac{1}{4}\,$

$Description: \int\frac{dx}{x^2-4}\,$

$Description: = \frac{1}{4} \int (\frac{1}{x-2} - \frac {1}{x+2})\,dx\,$

$Description: = \frac{1}{4} (ln|x-2| - ln|x+2|) + C\,$

$Description: = \frac{1}{4} ln|\frac{x-2}{x+2}| + C\,$

Rumus integrasi dasar

Umum

$Description: \int x^n\,dx = \frac{x^{n+1}}{n+1} + C\,$ (n ≠ -1)

$Description: \int\frac{d}{dx}[f(x)]\,dx = f(x) + C\,$

$Description: \int(u+v)\,dx = \int u \,dx + \int v \,dx\,$

$Description: \int au\,dx = a \int u\, dx\,$ (a adalah konstanta)

$Description: \int \frac{dx}{x} = ln |x| + C\,$

$Description: \int a^x \,dx = \frac{a^x}{ln a} + C\,$ (a > 0, a ≠ 1)

$Description: \int \frac{dx}{x} = ln |x| + C$

Bilangan natural

$Description: \int e^u du= e^u + C\,$

Logaritma

$Description: \int \log_b(x) \,dx = x \log_b(x) - \frac{x}{\ln(b)} + C = x \log_b \left(\frac{x}{e}\right) + C$

Trigonometri

$Description: \int\sin x\,dx = -\cos x + C\,$

$Description: \int\cos x\,dx = \sin x + C\,$

$Description: \int\tan x\,dx = \ln |\sec x| + C\,$

$Description: \int\cot x\,dx = \ln |\sin x| + C\,$

$Description: \int\sec x\,dx = \ln |\sec x + \tan x| + C\,$

$Description: \int\csc x\,dx = \ln |\csc x - \cot x| + C\,$

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

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

$Description: \int\sec x\tan x\,dx = \sec x + C\,$

$Description: \int\csc x\cot x\,dx = -\csc x + C\,$

Senin, 13 Agustus 2012