Integral de $$$\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$

Encontre $$$\int \cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\, dx$$$.

Para a integral $$$\int{\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}$$$ e $$$\operatorname{dv}=dx$$$.

Então $$$\operatorname{du}=\left(\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}\right)^{\prime }dx=- \frac{2 \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3 x} dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{1 d x}=x$$$ (os passos podem ser vistos »).

Portanto,

$${\color{red}{\int{\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x}}}={\color{red}{\left(\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} \cdot x-\int{x \cdot \left(- \frac{2 \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3 x}\right) d x}\right)}}={\color{red}{\left(x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} - \int{\left(- \frac{2 \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3}\right)d x}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=- \frac{2}{3}$$$ e $$$f{\left(x \right)} = \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}$$$:

$$x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} - {\color{red}{\int{\left(- \frac{2 \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3}\right)d x}}} = x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} - {\color{red}{\left(- \frac{2 \int{\sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x}}{3}\right)}}$$

Para a integral $$$\int{\sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=\sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}$$$ e $$$\operatorname{dv}=dx$$$.

Então $$$\operatorname{du}=\left(\sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}\right)^{\prime }dx=\frac{2 \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3 x} dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{1 d x}=x$$$ (os passos podem ser vistos »).

Assim,

$$x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} + \frac{2 {\color{red}{\int{\sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x}}}}{3}=x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} + \frac{2 {\color{red}{\left(\sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} \cdot x-\int{x \cdot \frac{2 \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3 x} d x}\right)}}}{3}=x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} + \frac{2 {\color{red}{\left(x \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} - \int{\frac{2 \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3} d x}\right)}}}{3}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{2}{3}$$$ e $$$f{\left(x \right)} = \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}$$$:

$$\frac{2 x \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3} + x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} - \frac{2 {\color{red}{\int{\frac{2 \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3} d x}}}}{3} = \frac{2 x \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3} + x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} - \frac{2 {\color{red}{\left(\frac{2 \int{\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x}}{3}\right)}}}{3}$$

Chegamos a uma integral que já vimos.

Assim, obtivemos a seguinte equação simples em relação à integral:

$$\int{\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x} = \frac{2 x \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}}{3} + x \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} - \frac{4 \int{\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x}}{9}$$

Resolvendo, obtemos que

$$\int{\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x} = \frac{3 x \left(2 \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} + 3 \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}\right)}{13}$$

Portanto,

$$\int{\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x} = \frac{3 x \left(2 \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} + 3 \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}\right)}{13}$$

Adicione a constante de integração:

$$\int{\cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} d x} = \frac{3 x \left(2 \sin{\left(\frac{2 \ln{\left(x \right)}}{3} \right)} + 3 \cos{\left(\frac{2 \ln{\left(x \right)}}{3} \right)}\right)}{13}+C$$

$$$\int \cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\, dx = \frac{3 x \left(2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} + 3 \cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right)}{13} + C$$$A