$$$\tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)}$$$ 的積分

求$$$\int \tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)}\, dx$$$。

令 $$$u=\tan{\left(x \right)}$$$。

則 $$$du=\left(\tan{\left(x \right)}\right)^{\prime }dx = \sec^{2}{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\sec^{2}{\left(x \right)} dx = du$$$。

該積分可改寫為

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

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：

$${\color{red}{\int{u^{2} d u}}}={\color{red}{\frac{u^{1 + 2}}{1 + 2}}}={\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

回顧一下 $$$u=\tan{\left(x \right)}$$$：

$$\frac{{\color{red}{u}}^{3}}{3} = \frac{{\color{red}{\tan{\left(x \right)}}}^{3}}{3}$$

因此，

$$\int{\tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)} d x} = \frac{\tan^{3}{\left(x \right)}}{3}$$

加上積分常數：

$$\int{\tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)} d x} = \frac{\tan^{3}{\left(x \right)}}{3}+C$$

$$$\int \tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)}\, dx = \frac{\tan^{3}{\left(x \right)}}{3} + C$$$A