Integral dari $$$\sqrt{2} x^{\frac{5}{2}}$$$

Temukan $$$\int \sqrt{2} x^{\frac{5}{2}}\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=\sqrt{2}$$$ dan $$$f{\left(x \right)} = x^{\frac{5}{2}}$$$:

$${\color{red}{\int{\sqrt{2} x^{\frac{5}{2}} d x}}} = {\color{red}{\sqrt{2} \int{x^{\frac{5}{2}} d x}}}$$

Terapkan aturan pangkat $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ dengan $$$n=\frac{5}{2}$$$:

$$\sqrt{2} {\color{red}{\int{x^{\frac{5}{2}} d x}}}=\sqrt{2} {\color{red}{\frac{x^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=\sqrt{2} {\color{red}{\left(\frac{2 x^{\frac{7}{2}}}{7}\right)}}$$

Oleh karena itu,

$$\int{\sqrt{2} x^{\frac{5}{2}} d x} = \frac{2 \sqrt{2} x^{\frac{7}{2}}}{7}$$

Tambahkan konstanta integrasi:

$$\int{\sqrt{2} x^{\frac{5}{2}} d x} = \frac{2 \sqrt{2} x^{\frac{7}{2}}}{7}+C$$

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