Integral dengan Subtitusi ~ blajar-pintar

Thursday, March 15, 2012

Integral dengan Subtitusi

4:17 PM Smufy 0

Contoh 1:
Tentukan nilai dari
$\int6x(x^2+4)^2dx$

Penyelesaian:
Misalkan: U=x²+4 → du=2x dx
Subtitusikan du=2x dx ke persamaan awal, sehingga:

$\int6x(x^2+4)^2dx$	=	$\int3(2x)(x^2+4)^2dx$
	=	$\int 3U^2dU$
	=	U³+ C
	=	(x²+4)³+ C

Contoh 2:
Tentukan nilai dari
$\int x(x^2+2)^4dx$

Penyelesaian:
Misalkan: U=x²+2 → du=2x dx
Subtitusikan du=2x dx ke persamaan awal, sehingga:

$\int x(x^2+2)^4dx$	=	$\int\frac{1}{2}(2x)(x^2+2)^4dx$
	=	$\int \frac{1}{2}U^4dU$
	=	$\frac{1}{2}.\frac{1}{5}U^5+C$
	=	$\frac{1}{10}U^5+C$
	=	$\frac{1}{10}(x^2+2)^5+C$

Contoh 3:
Tentukan nilai dari integral berikut:
$\int_{0}^{\sqrt{5}}x\sqrt{x^2+4}dx$

Penyelesaian:
Misalkan: U=x²+4 → du=2x dx
untuk x=0 → U=4 dan x= $\sqrt{5}$ → U=9
Subtitusikan du=2x dx ke persamaan awal, sehingga:

$\int_{0}^{\sqrt{5}}x\sqrt{x^2+4}dx$	=	$\int_{4}^{9}\frac{1}{2}\sqrt{U}du$
	=	$\left[\frac{1}{2}.\frac{2}{3}U^{\frac{3}{2}}\right ]_{4}^{9}$
	=	$\left[\frac{1}{3}U\sqrt{U}\right ]_{4}^{9}$
	=	$\frac{1}{3}\left[9\sqrt{9}-4\sqrt{4}\right ]$
	=	$\frac{1}{3}\left[27-8\right ]$
	=	$\frac{19}{3}$

Latihan Soal:
$1.\int2x^2(x^4-1)^2dx$
$2. \int x^2\sqrt{x^3-2}dx$
$3. \int \frac{2x}{\sqrt{3x^2+1}}dx$
$4. \int(x^2-4x+6)^{\frac{2}{3}dx$
$5. \int x\sqrt[4]{(x^2-3)^2}dx$
$6.\int_{-1}^{0}x(x^2+3)^3dx$
$7. \int_{0}^{4}x\sqrt{2x+1}dx$
$8. \int_{0}^{2}\frac{x^2+2x}{x^3+3x^2+1}dx$
$9. \int_{0}^{1}\frac{x}{5x^2+4}dx$
$10. \int_{-1}^{1}4x^3(4x^4-1)^2dx$

Posted in: Integral, Matematika

Hidup Sehat

Thursday, March 15, 2012