sengguan.com: Integral (Contoh Soal dan Penyelesaianya)

Contoh 1 ada pada postingan Integral Substitusi

Contoh 2 :

$\int x\left ( x^{2}+8 \right )^{19}dx = ...$

Jawab :
Misal y = x² + 8
maka
$\frac{dy}{dx}=2x$
Sehingga
$dx=\frac{dy}{2x}$

Maka
$\int x\left ( x^{2}+8 \right )^{19}dx = \int xy^{19}\frac{dy}{2x}=\frac{1}{2}\int y^{19}dy$
$= \frac{1}{2}.\frac{1}{20}y^{20} + c = \frac{1}{40}\left ( x^{2} + 8 \right )^{20} + c$

Contoh 3 :
$\int x^{3}\left ( x^{4} - 12 \right )^{24} dx = ...$

Jawab :
misal y = x⁴ - 12
maka
$\frac{dy}{dx}= 4x^{3}$
Akibatnya
$dx=\frac{dy}{4x^{3}}$
jadi

$\int x^{3}\left ( x^{4}-12 \right )^{24}dx = \int x^{3}y^{24}\frac{dy}{4x^{3}}=\frac{1}{4}\int y^{24}dy$
$\frac{1}{4}.\frac{1}{25}y^{25}+ c = \frac{1}{100}\left ( x^{4} - 12 \right ) + c$

Contoh 4
Hitunglah Integral Berikut
$\int \left ( x+3 \right )\left ( x^{2}+6x+5 \right )^{12}dx=...$

Jawab :
misal y = x² + 6x + 5
maka
$\frac{dy}{dx}=2x+6$
sehingga
$dx=\frac{dy}{2x+6}=\frac{dy}{2(x+3)}$
Jadi :
$\int \left ( x+3 \right )\left ( x^{2}+6x+5 \right )^{12}dx =\int \left ( x+3 \right )y^{12}\frac{dy}{2(x+3)}=\frac{1}{2}\int y^{12}dy$

$=\frac{1}{2}.\frac{1}{13}y^{13}+c=\frac{1}{26}(x^{2}+6x+5)^{13}+c$

Contoh 5 :
Carilah hasil integral berikut
$\int (x^{2}-1)\sqrt{x^{3}-3x+5}dx=...$

Jawab :
misal : y = x³-3x+5
sehingga
$\frac{dy}{dx}=3x^{2}-3$
maka
$dx=\frac{dy}{3x^{2}-3}=\frac{dy}{3(x^{2}-1)}$
Dengan demikian
$\int (x^{2}-1)\sqrt{x^{3}-3x+5}dx = \int (x^{2}-1)\sqrt{y}.\frac{dy}{3(x^{2}-1)}=\frac{1}{3}\int y^{\frac{1}{2}}dy$
$=\frac{1}{3}.\frac{2}{3}y^{\frac{3}{2}}+c=\frac{2}{9}\left ( x^{3}-3x+5 \right )^{\frac{3}{2}}+c$

Sumber : http://web-matematika.blogspot.com/