$$$\frac{\sqrt{x^{2} - 1}}{x - 1}$$$ 的积分

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

设$$$x=\cosh{\left(u \right)}$$$。

则$$$dx=\left(\cosh{\left(u \right)}\right)^{\prime }du = \sinh{\left(u \right)} du$$$（步骤见»）。

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

所以，

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

利用恒等式 $$$\cosh^{2}{\left( u \right)} - 1 = \sinh^{2}{\left( u \right)}$$$：

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

假设$$$\sinh{\left( u \right)} \ge 0$$$，我们得到如下结果：

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

因此，

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

将双曲正弦用双曲余弦表示，进一步将分子改写，使用平方差公式，并化简。:

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

逐项积分：

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

应用常数法则 $$$\int c\, du = c u$$$，使用 $$$c=1$$$：

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

双曲余弦的积分为 $$$\int{\cosh{\left(u \right)} d u} = \sinh{\left(u \right)}$$$：

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

回忆一下 $$$u=\operatorname{acosh}{\left(x \right)}$$$:

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

因此，

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

加上积分常数：

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

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