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

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

Reescreva a secante como $$$\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$$$:

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

Reescreva o cosseno em termos do seno usando a fórmula $$$\cos\left(\theta\right)=\sin\left(\theta + \frac{\pi}{2}\right)$$$ e depois reescreva o seno usando a fórmula do ângulo duplo $$$\sin\left(\theta\right)=2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)$$$:

$${\color{red}{\int{\frac{1}{\cos{\left(\theta \right)}} d \theta}}} = {\color{red}{\int{\frac{1}{2 \sin{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}} d \theta}}}$$

Multiplique o numerador e o denominador por $$$\sec^2\left(\frac{\theta}{2} + \frac{\pi}{4} \right)$$$:

$${\color{red}{\int{\frac{1}{2 \sin{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}} d \theta}}} = {\color{red}{\int{\frac{\sec^{2}{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}}{2 \tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}} d \theta}}}$$

Seja $$$u=\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}$$$.

Então $$$du=\left(\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}\right)^{\prime }d\theta = \frac{\sec^{2}{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}}{2} d\theta$$$ (veja os passos »), e obtemos $$$\sec^{2}{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)} d\theta = 2 du$$$.

Portanto,

$${\color{red}{\int{\frac{\sec^{2}{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}}{2 \tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}} d \theta}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$

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 $$$u=\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}}}}\right| \right)}$$

Portanto,

$$\int{\sec{\left(\theta \right)} d \theta} = \ln{\left(\left|{\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}}\right| \right)}$$

Adicione a constante de integração:

$$\int{\sec{\left(\theta \right)} d \theta} = \ln{\left(\left|{\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}}\right| \right)}+C$$

$$$\int \sec{\left(\theta \right)}\, d\theta = \ln\left(\left|{\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}}\right|\right) + C$$$A