Integral de $$$e^{- \frac{y}{t}}$$$ em relação a $$$y$$$

Encontre $$$\int e^{- \frac{y}{t}}\, dy$$$.

Seja $$$u=- \frac{y}{t}$$$.

Então $$$du=\left(- \frac{y}{t}\right)^{\prime }dy = - \frac{1}{t} dy$$$ (veja os passos »), e obtemos $$$dy = - t du$$$.

Portanto,

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

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

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

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

$$- t {\color{red}{\int{e^{u} d u}}} = - t {\color{red}{e^{u}}}$$

Recorde que $$$u=- \frac{y}{t}$$$:

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

Portanto,

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

Adicione a constante de integração:

$$\int{e^{- \frac{y}{t}} d y} = - t e^{- \frac{y}{t}}+C$$

$$$\int e^{- \frac{y}{t}}\, dy = - t e^{- \frac{y}{t}} + C$$$A