Integral de $$$\frac{e^{- y}}{y}$$$

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

Seja $$$u=- y$$$.

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

Assim,

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

Esta integral (Integral Exponencial) não possui forma fechada:

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

Recorde que $$$u=- y$$$:

$$\operatorname{Ei}{\left({\color{red}{u}} \right)} = \operatorname{Ei}{\left({\color{red}{\left(- y\right)}} \right)}$$

Portanto,

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

Adicione a constante de integração:

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

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