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

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

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

Então $$$du=\left(x^{4}\right)^{\prime }dx = 4 x^{3} dx$$$ (veja os passos »), e obtemos $$$x^{3} dx = \frac{du}{4}$$$.

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{4}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

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

A integral da função exponencial é $$$\int{e^{u} d u} = e^{u}$$$:

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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