$$$\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}$$$ 的積分
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求$$$\int \frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}\, dx$$$。
解答
重寫被積函數:
$${\color{red}{\int{\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} d x}}} = {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}}$$
令 $$$u=\sec{\left(x \right)}$$$。
則 $$$du=\left(\sec{\left(x \right)}\right)^{\prime }dx = \tan{\left(x \right)} \sec{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\tan{\left(x \right)} \sec{\left(x \right)} dx = du$$$。
該積分變為
$${\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = {\color{red}{\int{1 d u}}}$$
配合 $$$c=1$$$，應用常數法則 $$$\int c\, du = c u$$$：
$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$
回顧一下 $$$u=\sec{\left(x \right)}$$$：
$${\color{red}{u}} = {\color{red}{\sec{\left(x \right)}}}$$
因此，
$$\int{\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} d x} = \sec{\left(x \right)}$$
加上積分常數：
$$\int{\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} d x} = \sec{\left(x \right)}+C$$
答案
$$$\int \frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}\, dx = \sec{\left(x \right)} + C$$$A