Tentukan hasil pengintegralan integral

Hasil pengintegralan adalah $\frac{2}{9}x^{9/2} + \frac{4}{7}x^{7/2} + \frac{2}{5}x^{5/2} + C$

Pembahasan

Untuk menentukan hasil pengintegralan dari $\frac{x^2(x+1)^2}{\sqrt{x}} dx$, kita perlu menyederhanakan ekspresi terlebih dahulu:

$\frac{x^2(x+1)^2}{\sqrt{x}} = \frac{x^2(x^2+2x+1)}{x^{1/2}}$

$= \frac{x^4+2x^3+x^2}{x^{1/2}}$

$= x^{4-1/2} + 2x^{3-1/2} + x^{2-1/2}$

$= x^{7/2} + 2x^{5/2} + x^{3/2}$

Sekarang kita dapat mengintegralkan setiap suku:

$\int x^{7/2} dx = \frac{x^{7/2+1}}{7/2+1} + C_1 = \frac{x^{9/2}}{9/2} + C_1 = \frac{2}{9}x^{9/2} + C_1$

$\int 2x^{5/2} dx = 2\frac{x^{5/2+1}}{5/2+1} + C_2 = 2\frac{x^{7/2}}{7/2} + C_2 = 2 \times \frac{2}{7}x^{7/2} + C_2 = \frac{4}{7}x^{7/2} + C_2$

$\int x^{3/2} dx = \frac{x^{3/2+1}}{3/2+1} + C_3 = \frac{x^{5/2}}{5/2} + C_3 = \frac{2}{5}x^{5/2} + C_3$

Jadi, hasil pengintegralan keseluruhannya adalah:

$\int \frac{x^2(x+1)^2}{\sqrt{x}} dx = \frac{2}{9}x^{9/2} + \frac{4}{7}x^{7/2} + \frac{2}{5}x^{5/2} + C$

Di mana C adalah konstanta integrasi.