$$$\frac{1}{f \cos{\left(x \right)}}$$$ 對 $$$x$$$ 的積分

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

使用公式 $$$\cos\left(x\right)=\sin\left(x + \frac{\pi}{2}\right)$$$ 將餘弦用正弦表示，然後使用二倍角公式 $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$ 將正弦改寫。:

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

將分子與分母同時乘以 $$$\sec^2\left(\frac{x}{2} + \frac{\pi}{4} \right)$$$:

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

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

則 $$$du=\left(\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{2} dx$$$ (步驟見»)，並可得 $$$\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} dx = 2 du$$$。

所以，

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{f}$$$ 與 $$$f{\left(u \right)} = \frac{1}{u}$$$：

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

$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：

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

回顧一下 $$$u=\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$$：

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

因此，

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

加上積分常數：

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

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