Integral de $$$2 x^{3} \sqrt{x^{5}}$$$

Halla $$$\int 2 x^{3} \sqrt{x^{5}}\, dx$$$.

La entrada se reescribe: $$$\int{2 x^{3} \sqrt{x^{5}} d x}=\int{2 x^{\frac{11}{2}} d x}$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=2$$$ y $$$f{\left(x \right)} = x^{\frac{11}{2}}$$$:

$${\color{red}{\int{2 x^{\frac{11}{2}} d x}}} = {\color{red}{\left(2 \int{x^{\frac{11}{2}} d x}\right)}}$$

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=\frac{11}{2}$$$:

$$2 {\color{red}{\int{x^{\frac{11}{2}} d x}}}=2 {\color{red}{\frac{x^{1 + \frac{11}{2}}}{1 + \frac{11}{2}}}}=2 {\color{red}{\left(\frac{2 x^{\frac{13}{2}}}{13}\right)}}$$

Por lo tanto,

$$\int{2 x^{\frac{11}{2}} d x} = \frac{4 x^{\frac{13}{2}}}{13}$$

Añade la constante de integración:

$$\int{2 x^{\frac{11}{2}} d x} = \frac{4 x^{\frac{13}{2}}}{13}+C$$

$$$\int 2 x^{3} \sqrt{x^{5}}\, dx = \frac{4 x^{\frac{13}{2}}}{13} + C$$$A