Integral de $$$4 \sqrt{5} x^{\frac{5}{2}}$$$

Halla $$$\int 4 \sqrt{5} x^{\frac{5}{2}}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=4 \sqrt{5}$$$ y $$$f{\left(x \right)} = x^{\frac{5}{2}}$$$:

$${\color{red}{\int{4 \sqrt{5} x^{\frac{5}{2}} d x}}} = {\color{red}{\left(4 \sqrt{5} \int{x^{\frac{5}{2}} d x}\right)}}$$

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=\frac{5}{2}$$$:

$$4 \sqrt{5} {\color{red}{\int{x^{\frac{5}{2}} d x}}}=4 \sqrt{5} {\color{red}{\frac{x^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=4 \sqrt{5} {\color{red}{\left(\frac{2 x^{\frac{7}{2}}}{7}\right)}}$$

Por lo tanto,

$$\int{4 \sqrt{5} x^{\frac{5}{2}} d x} = \frac{8 \sqrt{5} x^{\frac{7}{2}}}{7}$$

Añade la constante de integración:

$$\int{4 \sqrt{5} x^{\frac{5}{2}} d x} = \frac{8 \sqrt{5} x^{\frac{7}{2}}}{7}+C$$

$$$\int 4 \sqrt{5} x^{\frac{5}{2}}\, dx = \frac{8 \sqrt{5} x^{\frac{7}{2}}}{7} + C$$$A