Integral de $$$x^{3} e^{x^{4}}$$$

Halla $$$\int x^{3} e^{x^{4}}\, dx$$$.

Sea $$$u=x^{4}$$$.

Entonces $$$du=\left(x^{4}\right)^{\prime }dx = 4 x^{3} dx$$$ (los pasos pueden verse »), y obtenemos que $$$x^{3} dx = \frac{du}{4}$$$.

La integral se convierte en

$${\color{red}{\int{x^{3} e^{x^{4}} d x}}} = {\color{red}{\int{\frac{e^{u}}{4} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{4}$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\frac{e^{u}}{4} d u}}} = {\color{red}{\left(\frac{\int{e^{u} d u}}{4}\right)}}$$

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

$$\frac{{\color{red}{\int{e^{u} d u}}}}{4} = \frac{{\color{red}{e^{u}}}}{4}$$

Recordemos que $$$u=x^{4}$$$:

$$\frac{e^{{\color{red}{u}}}}{4} = \frac{e^{{\color{red}{x^{4}}}}}{4}$$

Por lo tanto,

$$\int{x^{3} e^{x^{4}} d x} = \frac{e^{x^{4}}}{4}$$

Añade la constante de integración:

$$\int{x^{3} e^{x^{4}} d x} = \frac{e^{x^{4}}}{4}+C$$

$$$\int x^{3} e^{x^{4}}\, dx = \frac{e^{x^{4}}}{4} + C$$$A