Integral de $$$x - 8 - \frac{4 e^{x}}{x}$$$

Encontre $$$\int \left(x - 8 - \frac{4 e^{x}}{x}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(x - 8 - \frac{4 e^{x}}{x}\right)d x}}} = {\color{red}{\left(- \int{8 d x} + \int{x d x} - \int{\frac{4 e^{x}}{x} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=8$$$:

$$\int{x d x} - \int{\frac{4 e^{x}}{x} d x} - {\color{red}{\int{8 d x}}} = \int{x d x} - \int{\frac{4 e^{x}}{x} d x} - {\color{red}{\left(8 x\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=1$$$:

$$- 8 x - \int{\frac{4 e^{x}}{x} d x} + {\color{red}{\int{x d x}}}=- 8 x - \int{\frac{4 e^{x}}{x} d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 8 x - \int{\frac{4 e^{x}}{x} d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=4$$$ e $$$f{\left(x \right)} = \frac{e^{x}}{x}$$$:

$$\frac{x^{2}}{2} - 8 x - {\color{red}{\int{\frac{4 e^{x}}{x} d x}}} = \frac{x^{2}}{2} - 8 x - {\color{red}{\left(4 \int{\frac{e^{x}}{x} d x}\right)}}$$

Esta integral (Integral Exponencial) não possui forma fechada:

$$\frac{x^{2}}{2} - 8 x - 4 {\color{red}{\int{\frac{e^{x}}{x} d x}}} = \frac{x^{2}}{2} - 8 x - 4 {\color{red}{\operatorname{Ei}{\left(x \right)}}}$$

Portanto,

$$\int{\left(x - 8 - \frac{4 e^{x}}{x}\right)d x} = \frac{x^{2}}{2} - 8 x - 4 \operatorname{Ei}{\left(x \right)}$$

Adicione a constante de integração:

$$\int{\left(x - 8 - \frac{4 e^{x}}{x}\right)d x} = \frac{x^{2}}{2} - 8 x - 4 \operatorname{Ei}{\left(x \right)}+C$$

$$$\int \left(x - 8 - \frac{4 e^{x}}{x}\right)\, dx = \left(\frac{x^{2}}{2} - 8 x - 4 \operatorname{Ei}{\left(x \right)}\right) + C$$$A