Integral de $$$\frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}}$$$

Encontre $$$\int \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}}\, dx$$$.

Seja $$$u=\frac{1}{x}$$$.

Então $$$du=\left(\frac{1}{x}\right)^{\prime }dx = - \frac{1}{x^{2}} dx$$$ (veja os passos »), e obtemos $$$\frac{dx}{x^{2}} = - du$$$.

A integral torna-se

$${\color{red}{\int{\frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}} d x}}} = {\color{red}{\int{\left(- u \cos{\left(u \right)}\right)d u}}}$$

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

$${\color{red}{\int{\left(- u \cos{\left(u \right)}\right)d u}}} = {\color{red}{\left(- \int{u \cos{\left(u \right)} d u}\right)}}$$

Para a integral $$$\int{u \cos{\left(u \right)} d u}$$$, use integração por partes $$$\int \operatorname{\kappa} \operatorname{dv} = \operatorname{\kappa}\operatorname{v} - \int \operatorname{v} \operatorname{d\kappa}$$$.

Sejam $$$\operatorname{\kappa}=u$$$ e $$$\operatorname{dv}=\cos{\left(u \right)} du$$$.

Então $$$\operatorname{d\kappa}=\left(u\right)^{\prime }du=1 du$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{\cos{\left(u \right)} d u}=\sin{\left(u \right)}$$$ (os passos podem ser vistos »).

Assim,

$$- {\color{red}{\int{u \cos{\left(u \right)} d u}}}=- {\color{red}{\left(u \cdot \sin{\left(u \right)}-\int{\sin{\left(u \right)} \cdot 1 d u}\right)}}=- {\color{red}{\left(u \sin{\left(u \right)} - \int{\sin{\left(u \right)} d u}\right)}}$$

A integral do seno é $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$- u \sin{\left(u \right)} + {\color{red}{\int{\sin{\left(u \right)} d u}}} = - u \sin{\left(u \right)} + {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$

Recorde que $$$u=\frac{1}{x}$$$:

$$- \cos{\left({\color{red}{u}} \right)} - {\color{red}{u}} \sin{\left({\color{red}{u}} \right)} = - \cos{\left({\color{red}{\frac{1}{x}}} \right)} - {\color{red}{\frac{1}{x}}} \sin{\left({\color{red}{\frac{1}{x}}} \right)}$$

Portanto,

$$\int{\frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}} d x} = - \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}$$

Adicione a constante de integração:

$$\int{\frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}} d x} = - \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}+C$$

$$$\int \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}}\, dx = \left(- \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}\right) + C$$$A