Integral de $$$\frac{\sqrt{21} \sqrt{x^{3}}}{21}$$$

Encontre $$$\int \frac{\sqrt{21} \sqrt{x^{3}}}{21}\, dx$$$.

A entrada é reescrita como: $$$\int{\frac{\sqrt{21} \sqrt{x^{3}}}{21} d x}=\int{\frac{\sqrt{21} x^{\frac{3}{2}}}{21} d x}$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{\sqrt{21}}{21}$$$ e $$$f{\left(x \right)} = x^{\frac{3}{2}}$$$:

$${\color{red}{\int{\frac{\sqrt{21} x^{\frac{3}{2}}}{21} d x}}} = {\color{red}{\left(\frac{\sqrt{21} \int{x^{\frac{3}{2}} d x}}{21}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=\frac{3}{2}$$$:

$$\frac{\sqrt{21} {\color{red}{\int{x^{\frac{3}{2}} d x}}}}{21}=\frac{\sqrt{21} {\color{red}{\frac{x^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}}{21}=\frac{\sqrt{21} {\color{red}{\left(\frac{2 x^{\frac{5}{2}}}{5}\right)}}}{21}$$

Portanto,

$$\int{\frac{\sqrt{21} x^{\frac{3}{2}}}{21} d x} = \frac{2 \sqrt{21} x^{\frac{5}{2}}}{105}$$

Adicione a constante de integração:

$$\int{\frac{\sqrt{21} x^{\frac{3}{2}}}{21} d x} = \frac{2 \sqrt{21} x^{\frac{5}{2}}}{105}+C$$

$$$\int \frac{\sqrt{21} \sqrt{x^{3}}}{21}\, dx = \frac{2 \sqrt{21} x^{\frac{5}{2}}}{105} + C$$$A