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

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

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

Então $$$du=\left(x^{3}\right)^{\prime }dx = 3 x^{2} dx$$$ (veja os passos »), e obtemos $$$x^{2} dx = \frac{du}{3}$$$.

A integral pode ser reescrita como

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{3}$$$ e $$$f{\left(u \right)} = e^{u}$$$:

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

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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