Integral de $$$y^{3} e^{\frac{y^{2}}{2}}$$$

Encontre $$$\int y^{3} e^{\frac{y^{2}}{2}}\, dy$$$.

Seja $$$u=y^{2}$$$.

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

Portanto,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(u \right)} = u e^{\frac{u}{2}}$$$:

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

Para a integral $$$\int{u e^{\frac{u}{2}} d u}$$$, use integração por partes $$$\int \operatorname{\mu} \operatorname{dv} = \operatorname{\mu}\operatorname{v} - \int \operatorname{v} \operatorname{d\mu}$$$.

Sejam $$$\operatorname{\mu}=u$$$ e $$$\operatorname{dv}=e^{\frac{u}{2}} du$$$.

Então $$$\operatorname{d\mu}=\left(u\right)^{\prime }du=1 du$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{e^{\frac{u}{2}} d u}=2 e^{\frac{u}{2}}$$$ (os passos podem ser vistos »).

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=2$$$ e $$$f{\left(u \right)} = e^{\frac{u}{2}}$$$:

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

Seja $$$v=\frac{u}{2}$$$.

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

Assim,

$$u e^{\frac{u}{2}} - {\color{red}{\int{e^{\frac{u}{2}} d u}}} = u e^{\frac{u}{2}} - {\color{red}{\int{2 e^{v} d v}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ usando $$$c=2$$$ e $$$f{\left(v \right)} = e^{v}$$$:

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

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

$$u e^{\frac{u}{2}} - 2 {\color{red}{\int{e^{v} d v}}} = u e^{\frac{u}{2}} - 2 {\color{red}{e^{v}}}$$

Recorde que $$$v=\frac{u}{2}$$$:

$$u e^{\frac{u}{2}} - 2 e^{{\color{red}{v}}} = u e^{\frac{u}{2}} - 2 e^{{\color{red}{\left(\frac{u}{2}\right)}}}$$

Recorde que $$$u=y^{2}$$$:

$$- 2 e^{\frac{{\color{red}{u}}}{2}} + {\color{red}{u}} e^{\frac{{\color{red}{u}}}{2}} = - 2 e^{\frac{{\color{red}{y^{2}}}}{2}} + {\color{red}{y^{2}}} e^{\frac{{\color{red}{y^{2}}}}{2}}$$

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

$$\int{y^{3} e^{\frac{y^{2}}{2}} d y} = \left(y^{2} - 2\right) e^{\frac{y^{2}}{2}}+C$$

$$$\int y^{3} e^{\frac{y^{2}}{2}}\, dy = \left(y^{2} - 2\right) e^{\frac{y^{2}}{2}} + C$$$A