Integral de $$$f \left(x + \frac{1}{x}\right)$$$ em relação a $$$x$$$

Encontre $$$\int f \left(x + \frac{1}{x}\right)\, dx$$$.

Expand the expression:

$${\color{red}{\int{f \left(x + \frac{1}{x}\right) d x}}} = {\color{red}{\int{\left(f x + \frac{f}{x}\right)d x}}}$$

Integre termo a termo:

$${\color{red}{\int{\left(f x + \frac{f}{x}\right)d x}}} = {\color{red}{\left(\int{\frac{f}{x} d x} + \int{f x d x}\right)}}$$

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

$$\int{\frac{f}{x} d x} + {\color{red}{\int{f x d x}}} = \int{\frac{f}{x} d x} + {\color{red}{f \int{x d x}}}$$

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

$$f {\color{red}{\int{x d x}}} + \int{\frac{f}{x} d x}=f {\color{red}{\frac{x^{1 + 1}}{1 + 1}}} + \int{\frac{f}{x} d x}=f {\color{red}{\left(\frac{x^{2}}{2}\right)}} + \int{\frac{f}{x} d x}$$

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

$$\frac{f x^{2}}{2} + {\color{red}{\int{\frac{f}{x} d x}}} = \frac{f x^{2}}{2} + {\color{red}{f \int{\frac{1}{x} d x}}}$$

A integral de $$$\frac{1}{x}$$$ é $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$\frac{f x^{2}}{2} + f {\color{red}{\int{\frac{1}{x} d x}}} = \frac{f x^{2}}{2} + f {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Portanto,

$$\int{f \left(x + \frac{1}{x}\right) d x} = \frac{f x^{2}}{2} + f \ln{\left(\left|{x}\right| \right)}$$

Simplifique:

$$\int{f \left(x + \frac{1}{x}\right) d x} = \frac{f \left(x^{2} + 2 \ln{\left(\left|{x}\right| \right)}\right)}{2}$$

Adicione a constante de integração:

$$\int{f \left(x + \frac{1}{x}\right) d x} = \frac{f \left(x^{2} + 2 \ln{\left(\left|{x}\right| \right)}\right)}{2}+C$$

$$$\int f \left(x + \frac{1}{x}\right)\, dx = \frac{f \left(x^{2} + 2 \ln\left(\left|{x}\right|\right)\right)}{2} + C$$$A