$$$9 \tan^{2}{\left(x \right)}$$$ 的積分

求$$$\int 9 \tan^{2}{\left(x \right)}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=9$$$ 與 $$$f{\left(x \right)} = \tan^{2}{\left(x \right)}$$$：

$${\color{red}{\int{9 \tan^{2}{\left(x \right)} d x}}} = {\color{red}{\left(9 \int{\tan^{2}{\left(x \right)} d x}\right)}}$$

令 $$$u=\tan{\left(x \right)}$$$。

則 $$$x=\operatorname{atan}{\left(u \right)}$$$ 與 $$$dx=\left(\operatorname{atan}{\left(u \right)}\right)^{\prime }du = \frac{du}{u^{2} + 1}$$$（步驟見»）。

該積分可改寫為

$$9 {\color{red}{\int{\tan^{2}{\left(x \right)} d x}}} = 9 {\color{red}{\int{\frac{u^{2}}{u^{2} + 1} d u}}}$$

重寫並拆分分式:

$$9 {\color{red}{\int{\frac{u^{2}}{u^{2} + 1} d u}}} = 9 {\color{red}{\int{\left(1 - \frac{1}{u^{2} + 1}\right)d u}}}$$

逐項積分：

$$9 {\color{red}{\int{\left(1 - \frac{1}{u^{2} + 1}\right)d u}}} = 9 {\color{red}{\left(\int{1 d u} - \int{\frac{1}{u^{2} + 1} d u}\right)}}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, du = c u$$$：

$$- 9 \int{\frac{1}{u^{2} + 1} d u} + 9 {\color{red}{\int{1 d u}}} = - 9 \int{\frac{1}{u^{2} + 1} d u} + 9 {\color{red}{u}}$$

$$$\frac{1}{u^{2} + 1}$$$ 的積分是 $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$：

$$9 u - 9 {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}} = 9 u - 9 {\color{red}{\operatorname{atan}{\left(u \right)}}}$$

回顧一下 $$$u=\tan{\left(x \right)}$$$：

$$- 9 \operatorname{atan}{\left({\color{red}{u}} \right)} + 9 {\color{red}{u}} = - 9 \operatorname{atan}{\left({\color{red}{\tan{\left(x \right)}}} \right)} + 9 {\color{red}{\tan{\left(x \right)}}}$$

因此，

$$\int{9 \tan^{2}{\left(x \right)} d x} = 9 \tan{\left(x \right)} - 9 \operatorname{atan}{\left(\tan{\left(x \right)} \right)}$$

化簡：

$$\int{9 \tan^{2}{\left(x \right)} d x} = - 9 x + 9 \tan{\left(x \right)}$$

加上積分常數：

$$\int{9 \tan^{2}{\left(x \right)} d x} = - 9 x + 9 \tan{\left(x \right)}+C$$

$$$\int 9 \tan^{2}{\left(x \right)}\, dx = \left(- 9 x + 9 \tan{\left(x \right)}\right) + C$$$A