Integral de $$$\frac{1}{f \cos{\left(x \right)}}$$$ em relação a $$$x$$$

Encontre $$$\int \frac{1}{f \cos{\left(x \right)}}\, dx$$$.

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

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

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

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

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

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

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{f}$$$ e $$$f{\left(u \right)} = \frac{1}{u}$$$:

$${\color{red}{\int{\frac{1}{f u} d u}}} = {\color{red}{\frac{\int{\frac{1}{u} d u}}{f}}}$$

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

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

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

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

Portanto,

$$\int{\frac{1}{f \cos{\left(x \right)}} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)}}{f}$$

Adicione a constante de integração:

$$\int{\frac{1}{f \cos{\left(x \right)}} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)}}{f}+C$$

$$$\int \frac{1}{f \cos{\left(x \right)}}\, dx = \frac{\ln\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right|\right)}{f} + C$$$A