$$$\operatorname{acosh}{\left(x \right)}$$$ 的积分

求$$$\int \operatorname{acosh}{\left(x \right)}\, dx$$$。

对于积分$$$\int{\operatorname{acosh}{\left(x \right)} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\operatorname{acosh}{\left(x \right)}$$$ 和 $$$\operatorname{dv}=dx$$$。

则 $$$\operatorname{du}=\left(\operatorname{acosh}{\left(x \right)}\right)^{\prime }dx=\frac{1}{\sqrt{x - 1} \sqrt{x + 1}} dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{1 d x}=x$$$ (步骤见 »)。

该积分可以改写为

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

设$$$u=x^{2} - 1$$$。

则$$$du=\left(x^{2} - 1\right)^{\prime }dx = 2 x dx$$$ (步骤见»)，并有$$$x dx = \frac{du}{2}$$$。

积分变为

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

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

应用幂法则 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=- \frac{1}{2}$$$：

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

回忆一下 $$$u=x^{2} - 1$$$:

$$x \operatorname{acosh}{\left(x \right)} - \sqrt{{\color{red}{u}}} = x \operatorname{acosh}{\left(x \right)} - \sqrt{{\color{red}{\left(x^{2} - 1\right)}}}$$

因此，

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

加上积分常数：

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

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