c. integral akar(x)(x+1/x) dx d. integral (x^3+x^2)/akar(x)

c. $\frac{2}{5}x^{5/2} + 2x^{1/2} + C$ d. $\frac{2}{7}x^{7/2} + \frac{2}{5}x^{5/2} + C$

Pembahasan

Untuk menyelesaikan integral-integral tersebut, kita akan menggunakan sifat-sifat integral dan aturan pangkat.

c. Integral dari $\sqrt{x}(x + \frac{1}{x}) dx$
Langkah 1: Distribusikan $\sqrt{x}$ ke dalam kurung.
$\sqrt{x}(x + \frac{1}{x}) = x^{1/2}(x^1 + x^{-1}) = x^{1/2} \cdot x^1 + x^{1/2} \cdot x^{-1} = x^{3/2} + x^{-1/2}$

Langkah 2: Integralkan setiap suku menggunakan aturan pangkat $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (untuk n ≠ -1).
$\int (x^{3/2} + x^{-1/2}) dx = \int x^{3/2} dx + \int x^{-1/2} dx$
$= \frac{x^{3/2 + 1}}{3/2 + 1} + \frac{x^{-1/2 + 1}}{-1/2 + 1} + C$
$= \frac{x^{5/2}}{5/2} + \frac{x^{1/2}}{1/2} + C$
$= \frac{2}{5}x^{5/2} + 2x^{1/2} + C$

Jadi, $\int \sqrt{x}(x + \frac{1}{x}) dx = \frac{2}{5}x^{5/2} + 2x^{1/2} + C$

d. Integral dari $\frac{x^3 + x^2}{\sqrt{x}} dx$
Langkah 1: Tulis ulang $\sqrt{x}$ sebagai $x^{1/2}$ dan bagi setiap suku di pembilang dengan $x^{1/2}$.
$\frac{x^3 + x^2}{x^{1/2}} = \frac{x^3}{x^{1/2}} + \frac{x^2}{x^{1/2}} = x^{3 - 1/2} + x^{2 - 1/2} = x^{5/2} + x^{3/2}$

Langkah 2: Integralkan setiap suku menggunakan aturan pangkat.
$\int (x^{5/2} + x^{3/2}) dx = \int x^{5/2} dx + \int x^{3/2} dx$
$= \frac{x^{5/2 + 1}}{5/2 + 1} + \frac{x^{3/2 + 1}}{3/2 + 1} + C$
$= \frac{x^{7/2}}{7/2} + \frac{x^{5/2}}{5/2} + C$
$= \frac{2}{7}x^{7/2} + \frac{2}{5}x^{5/2} + C$

Jadi, $\int \frac{x^3 + x^2}{\sqrt{x}} dx = \frac{2}{7}x^{7/2} + \frac{2}{5}x^{5/2} + C$