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

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

Simplifique o integrando:

$${\color{red}{\int{\frac{\sec{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{2}{\sin{\left(2 x \right)}} d x}}}$$

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

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

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

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

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

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

A integral pode ser reescrita como

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

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 $$$u=\tan{\left(x \right)}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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