$$$\tan{\left(x \right)} \sec^{p}{\left(x \right)}$$$ 對 $$$x$$$ 的積分

求$$$\int \tan{\left(x \right)} \sec^{p}{\left(x \right)}\, dx$$$。

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 $$$\tan{\left(x \right)} \sec{\left(x \right)} dx = du$$$。

該積分可改寫為

$${\color{red}{\int{\tan{\left(x \right)} \sec^{p}{\left(x \right)} d x}}} = {\color{red}{\int{u^{p - 1} d u}}}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=p - 1$$$：

$${\color{red}{\int{u^{p - 1} d u}}}={\color{red}{\frac{u^{\left(p - 1\right) + 1}}{\left(p - 1\right) + 1}}}={\color{red}{\frac{u^{p}}{p}}}$$

回顧一下 $$$u=\sec{\left(x \right)}$$$：

$$\frac{{\color{red}{u}}^{p}}{p} = \frac{{\color{red}{\sec{\left(x \right)}}}^{p}}{p}$$

因此，

$$\int{\tan{\left(x \right)} \sec^{p}{\left(x \right)} d x} = \frac{\sec^{p}{\left(x \right)}}{p}$$

加上積分常數：

$$\int{\tan{\left(x \right)} \sec^{p}{\left(x \right)} d x} = \frac{\sec^{p}{\left(x \right)}}{p}+C$$

$$$\int \tan{\left(x \right)} \sec^{p}{\left(x \right)}\, dx = \frac{\sec^{p}{\left(x \right)}}{p} + C$$$A