$$$\frac{\sqrt{x^{2} - 1}}{x - 1}$$$ 的積分

求$$$\int \frac{\sqrt{x^{2} - 1}}{x - 1}\, dx$$$。

令 $$$x=\cosh{\left(u \right)}$$$。

則 $$$dx=\left(\cosh{\left(u \right)}\right)^{\prime }du = \sinh{\left(u \right)} du$$$（步驟見»）。

此外，由此可得 $$$u=\operatorname{acosh}{\left(x \right)}$$$。

因此，

$$$\frac{\sqrt{x^{2} - 1}}{x - 1} = \frac{\sqrt{\cosh^{2}{\left( u \right)} - 1}}{\cosh{\left( u \right)} - 1}$$$

使用恆等式 $$$\cosh^{2}{\left( u \right)} - 1 = \sinh^{2}{\left( u \right)}$$$：

$$$\frac{\sqrt{\cosh^{2}{\left( u \right)} - 1}}{\cosh{\left( u \right)} - 1}=\frac{\sqrt{\sinh^{2}{\left( u \right)}}}{\cosh{\left( u \right)} - 1}$$$

假設 $$$\sinh{\left( u \right)} \ge 0$$$，可得如下：

$$$\frac{\sqrt{\sinh^{2}{\left( u \right)}}}{\cosh{\left( u \right)} - 1} = \frac{\sinh{\left( u \right)}}{\cosh{\left( u \right)} - 1}$$$

積分變為

$${\color{red}{\int{\frac{\sqrt{x^{2} - 1}}{x - 1} d x}}} = {\color{red}{\int{\frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)} - 1} d u}}}$$

以雙曲餘弦表示雙曲正弦，進一步改寫分子，使用平方差公式，並化簡。:

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

逐項積分：

$${\color{red}{\int{\left(\cosh{\left(u \right)} + 1\right)d u}}} = {\color{red}{\left(\int{1 d u} + \int{\cosh{\left(u \right)} d u}\right)}}$$

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

$$\int{\cosh{\left(u \right)} d u} + {\color{red}{\int{1 d u}}} = \int{\cosh{\left(u \right)} d u} + {\color{red}{u}}$$

雙曲餘弦的積分為 $$$\int{\cosh{\left(u \right)} d u} = \sinh{\left(u \right)}$$$：

$$u + {\color{red}{\int{\cosh{\left(u \right)} d u}}} = u + {\color{red}{\sinh{\left(u \right)}}}$$

回顧一下 $$$u=\operatorname{acosh}{\left(x \right)}$$$：

$$\sinh{\left({\color{red}{u}} \right)} + {\color{red}{u}} = \sinh{\left({\color{red}{\operatorname{acosh}{\left(x \right)}}} \right)} + {\color{red}{\operatorname{acosh}{\left(x \right)}}}$$

因此，

$$\int{\frac{\sqrt{x^{2} - 1}}{x - 1} d x} = \sqrt{x - 1} \sqrt{x + 1} + \operatorname{acosh}{\left(x \right)}$$

加上積分常數：

$$\int{\frac{\sqrt{x^{2} - 1}}{x - 1} d x} = \sqrt{x - 1} \sqrt{x + 1} + \operatorname{acosh}{\left(x \right)}+C$$

$$$\int \frac{\sqrt{x^{2} - 1}}{x - 1}\, dx = \left(\sqrt{x - 1} \sqrt{x + 1} + \operatorname{acosh}{\left(x \right)}\right) + C$$$A