Integral de $$$x^{2} \left(x^{3} - 1\right)^{10}$$$

Halla $$$\int x^{2} \left(x^{3} - 1\right)^{10}\, dx$$$.

Sea $$$u=x^{3} - 1$$$.

Entonces $$$du=\left(x^{3} - 1\right)^{\prime }dx = 3 x^{2} dx$$$ (los pasos pueden verse »), y obtenemos que $$$x^{2} dx = \frac{du}{3}$$$.

Por lo tanto,

$${\color{red}{\int{x^{2} \left(x^{3} - 1\right)^{10} d x}}} = {\color{red}{\int{\frac{u^{10}}{3} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{3}$$$ y $$$f{\left(u \right)} = u^{10}$$$:

$${\color{red}{\int{\frac{u^{10}}{3} d u}}} = {\color{red}{\left(\frac{\int{u^{10} d u}}{3}\right)}}$$

Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=10$$$:

$$\frac{{\color{red}{\int{u^{10} d u}}}}{3}=\frac{{\color{red}{\frac{u^{1 + 10}}{1 + 10}}}}{3}=\frac{{\color{red}{\left(\frac{u^{11}}{11}\right)}}}{3}$$

Recordemos que $$$u=x^{3} - 1$$$:

$$\frac{{\color{red}{u}}^{11}}{33} = \frac{{\color{red}{\left(x^{3} - 1\right)}}^{11}}{33}$$

Por lo tanto,

$$\int{x^{2} \left(x^{3} - 1\right)^{10} d x} = \frac{\left(x^{3} - 1\right)^{11}}{33}$$

Añade la constante de integración:

$$\int{x^{2} \left(x^{3} - 1\right)^{10} d x} = \frac{\left(x^{3} - 1\right)^{11}}{33}+C$$

$$$\int x^{2} \left(x^{3} - 1\right)^{10}\, dx = \frac{\left(x^{3} - 1\right)^{11}}{33} + C$$$A