$$$\frac{1}{f \cos{\left(x \right)}}$$$ 关于$$$x$$$的积分

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

使用公式$$$\cos\left(x\right)=\sin\left(x + \frac{\pi}{2}\right)$$$将余弦用正弦表示，然后使用二倍角公式$$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$将正弦改写。:

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

将分子和分母同时乘以 $$$\sec^2\left(\frac{x}{2} + \frac{\pi}{4} \right)$$$:

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

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

则$$$du=\left(\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{2} dx$$$ (步骤见»)，并有$$$\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} dx = 2 du$$$。

因此，

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

对 $$$c=\frac{1}{f}$$$ 和 $$$f{\left(u \right)} = \frac{1}{u}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

回忆一下 $$$u=\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$$:

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

因此，

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

加上积分常数：

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

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