Integral de $$$\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$$$

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

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

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

A integral pode ser reescrita como

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

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

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

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

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)}$$

Portanto,

$$\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)}$$

Adicione a constante de integração:

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

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