Integral de $$$3 e x^{2} \left(2 x^{3} - 8\right)$$$

Halla $$$\int 3 e x^{2} \left(2 x^{3} - 8\right)\, dx$$$.

Sea $$$u=2 x^{3} - 8$$$.

Entonces $$$du=\left(2 x^{3} - 8\right)^{\prime }dx = 6 x^{2} dx$$$ (los pasos pueden verse »), y obtenemos que $$$x^{2} dx = \frac{du}{6}$$$.

Por lo tanto,

$${\color{red}{\int{3 e x^{2} \left(2 x^{3} - 8\right) d x}}} = {\color{red}{\int{\frac{e u}{2} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{e}{2}$$$ y $$$f{\left(u \right)} = u$$$:

$${\color{red}{\int{\frac{e u}{2} d u}}} = {\color{red}{\left(\frac{e \int{u d u}}{2}\right)}}$$

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

$$\frac{e {\color{red}{\int{u d u}}}}{2}=\frac{e {\color{red}{\frac{u^{1 + 1}}{1 + 1}}}}{2}=\frac{e {\color{red}{\left(\frac{u^{2}}{2}\right)}}}{2}$$

Recordemos que $$$u=2 x^{3} - 8$$$:

$$\frac{e {\color{red}{u}}^{2}}{4} = \frac{e {\color{red}{\left(2 x^{3} - 8\right)}}^{2}}{4}$$

Por lo tanto,

$$\int{3 e x^{2} \left(2 x^{3} - 8\right) d x} = \frac{e \left(2 x^{3} - 8\right)^{2}}{4}$$

Simplificar:

$$\int{3 e x^{2} \left(2 x^{3} - 8\right) d x} = e \left(x^{3} - 4\right)^{2}$$

Añade la constante de integración:

$$\int{3 e x^{2} \left(2 x^{3} - 8\right) d x} = e \left(x^{3} - 4\right)^{2}+C$$

$$$\int 3 e x^{2} \left(2 x^{3} - 8\right)\, dx = e \left(x^{3} - 4\right)^{2} + C$$$A