Integral de $$$\cot{\left(\theta \right)}$$$

Encontre $$$\int \cot{\left(\theta \right)}\, d\theta$$$.

Reescreva a cotangente como $$$\cot\left(\theta\right)=\frac{\cos\left(\theta\right)}{\sin\left(\theta\right)}$$$:

$${\color{red}{\int{\cot{\left(\theta \right)} d \theta}}} = {\color{red}{\int{\frac{\cos{\left(\theta \right)}}{\sin{\left(\theta \right)}} d \theta}}}$$

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

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

A integral torna-se

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

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 $$$u=\sin{\left(\theta \right)}$$$:

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

Portanto,

$$\int{\cot{\left(\theta \right)} d \theta} = \ln{\left(\left|{\sin{\left(\theta \right)}}\right| \right)}$$

Adicione a constante de integração:

$$\int{\cot{\left(\theta \right)} d \theta} = \ln{\left(\left|{\sin{\left(\theta \right)}}\right| \right)}+C$$

$$$\int \cot{\left(\theta \right)}\, d\theta = \ln\left(\left|{\sin{\left(\theta \right)}}\right|\right) + C$$$A