Hasil dari integral ((2 x-1)(x+2))/(akar(x)) dx adalah ...

$\frac{4}{5} x^{5/2} + 2 x^{3/2} - 4 x^{1/2} + C$

Pembahasan

Untuk menyelesaikan integral $\\int \frac{(2x-1)(x+2)}{\sqrt{x}} dx$, pertama-tama kita perlu menyederhanakan fungsi di dalam integral.
Kalikan faktor-faktor di pembilang: $(2x-1)(x+2) = 2x^2 + 4x - x - 2 = 2x^2 + 3x - 2$.
Sekarang, bagi setiap suku dengan $\sqrt{x} = x^{1/2}$: $\frac{2x^2}{x^{1/2}} + \frac{3x}{x^{1/2}} - \frac{2}{x^{1/2}} = 2x^{2 - 1/2} + 3x^{1 - 1/2} - 2x^{-1/2} = 2x^{3/2} + 3x^{1/2} - 2x^{-1/2}$.
Sekarang, kita dapat mengintegralkan setiap suku menggunakan aturan pangkat $\\int x^n dx = \frac{x^{n+1}}{n+1} + C$:
$\\int (2x^{3/2} + 3x^{1/2} - 2x^{-1/2}) dx = 2 \int x^{3/2} dx + 3 \int x^{1/2} dx - 2 \int x^{-1/2} dx$
$= 2 \left( \frac{x^{3/2 + 1}}{3/2 + 1} \right) + 3 \left( \frac{x^{1/2 + 1}}{1/2 + 1} \right) - 2 \left( \frac{x^{-1/2 + 1}}{-1/2 + 1} \right) + C$
$= 2 \left( \frac{x^{5/2}}{5/2} \right) + 3 \left( \frac{x^{3/2}}{3/2} \right) - 2 \left( \frac{x^{1/2}}{1/2} \right) + C$
$= 2 \left( \frac{2}{5} x^{5/2} \right) + 3 \left( \frac{2}{3} x^{3/2} \right) - 2 (2 x^{1/2}) + C$
$= \frac{4}{5} x^{5/2} + 2 x^{3/2} - 4 x^{1/2} + C$.