$$$\frac{1}{\cos^{2}{\left(\theta \right)}}$$$ 的積分

求$$$\int \frac{1}{\cos^{2}{\left(\theta \right)}}\, d\theta$$$。

將被積函數以正割表示:

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

$$$\sec^{2}{\left(\theta \right)}$$$ 的積分是 $$$\int{\sec^{2}{\left(\theta \right)} d \theta} = \tan{\left(\theta \right)}$$$：

$${\color{red}{\int{\sec^{2}{\left(\theta \right)} d \theta}}} = {\color{red}{\tan{\left(\theta \right)}}}$$

因此，

$$\int{\frac{1}{\cos^{2}{\left(\theta \right)}} d \theta} = \tan{\left(\theta \right)}$$

加上積分常數：

$$\int{\frac{1}{\cos^{2}{\left(\theta \right)}} d \theta} = \tan{\left(\theta \right)}+C$$

$$$\int \frac{1}{\cos^{2}{\left(\theta \right)}}\, d\theta = \tan{\left(\theta \right)} + C$$$A