$$$\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}$$$ 的積分
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求$$$\int \frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}\, dx$$$。
解答
重寫被積函數:
$${\color{red}{\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x}}} = {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}}$$
$$$\sec^{2}{\left(x \right)}$$$ 的積分是 $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$：
$${\color{red}{\int{\sec^{2}{\left(x \right)} d x}}} = {\color{red}{\tan{\left(x \right)}}}$$
因此，
$$\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x} = \tan{\left(x \right)}$$
加上積分常數：
$$\int{\frac{\sec{\left(x \right)}}{\cos{\left(x \right)}} d x} = \tan{\left(x \right)}+C$$
答案
$$$\int \frac{\sec{\left(x \right)}}{\cos{\left(x \right)}}\, dx = \tan{\left(x \right)} + C$$$A