Integral de $$$\frac{\sqrt{2} \cos{\left(3 x \right)}}{4 \sin{\left(3 x \right)}}$$$

Encontre $$$\int \frac{\sqrt{2} \cos{\left(3 x \right)}}{4 \sin{\left(3 x \right)}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{\sqrt{2}}{4}$$$ e $$$f{\left(x \right)} = \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}$$$:

$${\color{red}{\int{\frac{\sqrt{2} \cos{\left(3 x \right)}}{4 \sin{\left(3 x \right)}} d x}}} = {\color{red}{\left(\frac{\sqrt{2} \int{\frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} d x}}{4}\right)}}$$

Seja $$$u=\sin{\left(3 x \right)}$$$.

Então $$$du=\left(\sin{\left(3 x \right)}\right)^{\prime }dx = 3 \cos{\left(3 x \right)} dx$$$ (veja os passos »), e obtemos $$$\cos{\left(3 x \right)} dx = \frac{du}{3}$$$.

A integral pode ser reescrita como

$$\frac{\sqrt{2} {\color{red}{\int{\frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} d x}}}}{4} = \frac{\sqrt{2} {\color{red}{\int{\frac{1}{3 u} d u}}}}{4}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{3}$$$ e $$$f{\left(u \right)} = \frac{1}{u}$$$:

$$\frac{\sqrt{2} {\color{red}{\int{\frac{1}{3 u} d u}}}}{4} = \frac{\sqrt{2} {\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{3}\right)}}}{4}$$

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

$$\frac{\sqrt{2} {\color{red}{\int{\frac{1}{u} d u}}}}{12} = \frac{\sqrt{2} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{12}$$

Recorde que $$$u=\sin{\left(3 x \right)}$$$:

$$\frac{\sqrt{2} \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{12} = \frac{\sqrt{2} \ln{\left(\left|{{\color{red}{\sin{\left(3 x \right)}}}}\right| \right)}}{12}$$

Portanto,

$$\int{\frac{\sqrt{2} \cos{\left(3 x \right)}}{4 \sin{\left(3 x \right)}} d x} = \frac{\sqrt{2} \ln{\left(\left|{\sin{\left(3 x \right)}}\right| \right)}}{12}$$

Adicione a constante de integração:

$$\int{\frac{\sqrt{2} \cos{\left(3 x \right)}}{4 \sin{\left(3 x \right)}} d x} = \frac{\sqrt{2} \ln{\left(\left|{\sin{\left(3 x \right)}}\right| \right)}}{12}+C$$

$$$\int \frac{\sqrt{2} \cos{\left(3 x \right)}}{4 \sin{\left(3 x \right)}}\, dx = \frac{\sqrt{2} \ln\left(\left|{\sin{\left(3 x \right)}}\right|\right)}{12} + C$$$A