$$$1 - \tan^{2}{\left(x \right)}$$$ 的積分
您的輸入
求$$$\int \left(1 - \tan^{2}{\left(x \right)}\right)\, dx$$$。
解答
逐項積分：
$${\color{red}{\int{\left(1 - \tan^{2}{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\tan^{2}{\left(x \right)} d x}\right)}}$$
配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：
$$- \int{\tan^{2}{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = - \int{\tan^{2}{\left(x \right)} d x} + {\color{red}{x}}$$
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»）。
因此，
$$x - {\color{red}{\int{\tan^{2}{\left(x \right)} d x}}} = x - {\color{red}{\int{\frac{u^{2}}{u^{2} + 1} d u}}}$$
重寫並拆分分式:
$$x - {\color{red}{\int{\frac{u^{2}}{u^{2} + 1} d u}}} = x - {\color{red}{\int{\left(1 - \frac{1}{u^{2} + 1}\right)d u}}}$$
逐項積分：
$$x - {\color{red}{\int{\left(1 - \frac{1}{u^{2} + 1}\right)d u}}} = x - {\color{red}{\left(\int{1 d u} - \int{\frac{1}{u^{2} + 1} d u}\right)}}$$
配合 $$$c=1$$$，應用常數法則 $$$\int c\, du = c u$$$：
$$x + \int{\frac{1}{u^{2} + 1} d u} - {\color{red}{\int{1 d u}}} = x + \int{\frac{1}{u^{2} + 1} d u} - {\color{red}{u}}$$
$$$\frac{1}{u^{2} + 1}$$$ 的積分是 $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$：
$$- u + x + {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}} = - u + x + {\color{red}{\operatorname{atan}{\left(u \right)}}}$$
回顧一下 $$$u=\tan{\left(x \right)}$$$：
$$x + \operatorname{atan}{\left({\color{red}{u}} \right)} - {\color{red}{u}} = x + \operatorname{atan}{\left({\color{red}{\tan{\left(x \right)}}} \right)} - {\color{red}{\tan{\left(x \right)}}}$$
因此，
$$\int{\left(1 - \tan^{2}{\left(x \right)}\right)d x} = x - \tan{\left(x \right)} + \operatorname{atan}{\left(\tan{\left(x \right)} \right)}$$
化簡：
$$\int{\left(1 - \tan^{2}{\left(x \right)}\right)d x} = 2 x - \tan{\left(x \right)}$$
加上積分常數：
$$\int{\left(1 - \tan^{2}{\left(x \right)}\right)d x} = 2 x - \tan{\left(x \right)}+C$$
答案
$$$\int \left(1 - \tan^{2}{\left(x \right)}\right)\, dx = \left(2 x - \tan{\left(x \right)}\right) + C$$$A