Integral de $$$x^{2} e^{x^{3}}$$$

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

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

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

La integral se convierte en

$${\color{red}{\int{x^{2} e^{x^{3}} d x}}} = {\color{red}{\int{\frac{e^{u}}{3} d u}}}$$

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 $$$f{\left(u \right)} = e^{u}$$$:

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

La integral de la función exponencial es $$$\int{e^{u} d u} = e^{u}$$$:

$$\frac{{\color{red}{\int{e^{u} d u}}}}{3} = \frac{{\color{red}{e^{u}}}}{3}$$

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

$$\frac{e^{{\color{red}{u}}}}{3} = \frac{e^{{\color{red}{x^{3}}}}}{3}$$

Por lo tanto,

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

Añade la constante de integración:

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

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