$$$\sec{\left(\theta \right)}$$$ 的积分

求$$$\int \sec{\left(\theta \right)}\, d\theta$$$。

将正割改写为 $$$\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$$$:

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

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

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

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

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

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

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

该积分可以改写为

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

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

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

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

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

因此，

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

加上积分常数：

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

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