Integral de $$$e^{\frac{2 y}{x}}$$$ em relação a $$$x$$$

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

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

Sejam $$$\operatorname{u}=e^{\frac{2 y}{x}}$$$ e $$$\operatorname{dv}=dx$$$.

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

Assim,

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

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

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

Seja $$$u=\frac{2 y}{x}$$$.

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

Assim,

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

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

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

Esta integral (Integral Exponencial) não possui forma fechada:

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

Recorde que $$$u=\frac{2 y}{x}$$$:

$$x e^{\frac{2 y}{x}} - 2 y \operatorname{Ei}{\left({\color{red}{u}} \right)} = x e^{\frac{2 y}{x}} - 2 y \operatorname{Ei}{\left({\color{red}{\left(\frac{2 y}{x}\right)}} \right)}$$

Portanto,

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

Adicione a constante de integração:

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

$$$\int e^{\frac{2 y}{x}}\, dx = \left(x e^{\frac{2 y}{x}} - 2 y \operatorname{Ei}{\left(\frac{2 y}{x} \right)}\right) + C$$$A