Integral de $$$\cot^{2}{\left(x \right)} \csc{\left(x \right)}$$$

Encontre $$$\int \cot^{2}{\left(x \right)} \csc{\left(x \right)}\, dx$$$.

Reescreva a tangente em função da cossecante, usando a fórmula $$$\cot^2\left(x \right)=\csc^2\left(x \right)-1$$$:

$${\color{red}{\int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x}}} = {\color{red}{\int{\left(\csc^{2}{\left(x \right)} - 1\right) \csc{\left(x \right)} d x}}}$$

Expanda a expressão:

$${\color{red}{\int{\left(\csc^{2}{\left(x \right)} - 1\right) \csc{\left(x \right)} d x}}} = {\color{red}{\int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}}}$$

Integre termo a termo:

$${\color{red}{\int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{\csc{\left(x \right)} d x} + \int{\csc^{3}{\left(x \right)} d x}\right)}}$$

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

Sejam $$$\operatorname{u}=\csc{\left(x \right)}$$$ e $$$\operatorname{dv}=\csc^{2}{\left(x \right)} dx$$$.

Então $$$\operatorname{du}=\left(\csc{\left(x \right)}\right)^{\prime }dx=- \cot{\left(x \right)} \csc{\left(x \right)} dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{\csc^{2}{\left(x \right)} d x}=- \cot{\left(x \right)}$$$ (os passos podem ser vistos »).

A integral pode ser reescrita como

$$\int{\csc^{3}{\left(x \right)} d x}=\csc{\left(x \right)} \cdot \left(- \cot{\left(x \right)}\right)-\int{\left(- \cot{\left(x \right)}\right) \cdot \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right) d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x}$$

Aplique a fórmula $$$\cot^{2}{\left(x \right)} = \csc^{2}{\left(x \right)} - 1$$$:

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{2}{\left(x \right)} - 1\right) \csc{\left(x \right)} d x}$$

Expandir:

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{2}{\left(x \right)} - 1\right) \csc{\left(x \right)} d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}$$

A integral de uma diferença é a diferença das integrais:

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}=- \cot{\left(x \right)} \csc{\left(x \right)} + \int{\csc{\left(x \right)} d x} - \int{\csc^{3}{\left(x \right)} d x}$$

Assim, obtemos a seguinte equação linear simples em relação à integral:

$${\color{red}{\int{\csc^{3}{\left(x \right)} d x}}}=- \cot{\left(x \right)} \csc{\left(x \right)} + \int{\csc{\left(x \right)} d x} - {\color{red}{\int{\csc^{3}{\left(x \right)} d x}}}$$

Resolvendo, obtemos que

$$\int{\csc^{3}{\left(x \right)} d x}=- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{\int{\csc{\left(x \right)} d x}}{2}$$

Portanto,

$$- \int{\csc{\left(x \right)} d x} + {\color{red}{\int{\csc^{3}{\left(x \right)} d x}}} = - \int{\csc{\left(x \right)} d x} + {\color{red}{\left(- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{\int{\csc{\left(x \right)} d x}}{2}\right)}}$$

Reescreva a cossecante como $$$\csc\left(x\right)=\frac{1}{\sin\left(x\right)}$$$:

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\csc{\left(x \right)} d x}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{\sin{\left(x \right)}} d x}}}}{2}$$

Reescreva o seno usando a fórmula do ângulo duplo $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$:

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{\sin{\left(x \right)}} d x}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} d x}}}}{2}$$

Multiplique o numerador e o denominador por $$$\sec^2\left(\frac{x}{2} \right)$$$:

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} d x}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}}}{2}$$

Seja $$$u=\tan{\left(\frac{x}{2} \right)}$$$.

Então $$$du=\left(\tan{\left(\frac{x}{2} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2} dx$$$ (veja os passos »), e obtemos $$$\sec^{2}{\left(\frac{x}{2} \right)} dx = 2 du$$$.

Logo,

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2}$$

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

$$- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2} = - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} - \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}$$

Recorde que $$$u=\tan{\left(\frac{x}{2} \right)}$$$:

$$- \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} = - \frac{\ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} \right)}}}}\right| \right)}}{2} - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2}$$

Portanto,

$$\int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x} = - \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}}{2} - \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2}$$

Simplifique:

$$\int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x} = - \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} + \cot{\left(x \right)} \csc{\left(x \right)}}{2}$$

Adicione a constante de integração:

$$\int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x} = - \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} + \cot{\left(x \right)} \csc{\left(x \right)}}{2}+C$$

$$$\int \cot^{2}{\left(x \right)} \csc{\left(x \right)}\, dx = - \frac{\ln\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right|\right) + \cot{\left(x \right)} \csc{\left(x \right)}}{2} + C$$$A