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

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

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 $$$\cos{\left(x \right)} dx = du$$$.

Assim,

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

Seja $$$v=\ln{\left(u \right)}$$$.

Então $$$dv=\left(\ln{\left(u \right)}\right)^{\prime }du = \frac{du}{u}$$$ (veja os passos »), e obtemos $$$\frac{du}{u} = dv$$$.

A integral pode ser reescrita como

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

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

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

Recorde que $$$v=\ln{\left(u \right)}$$$:

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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