Integral de $$$\sqrt{3} \sqrt{x}$$$

Halla $$$\int \sqrt{3} \sqrt{x}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\sqrt{3}$$$ y $$$f{\left(x \right)} = \sqrt{x}$$$:

$${\color{red}{\int{\sqrt{3} \sqrt{x} d x}}} = {\color{red}{\sqrt{3} \int{\sqrt{x} d x}}}$$

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=\frac{1}{2}$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

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