$$$\operatorname{acosh}{\left(x \right)}$$$ 的積分

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

對於積分 $$$\int{\operatorname{acosh}{\left(x \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

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

則 $$$\operatorname{du}=\left(\operatorname{acosh}{\left(x \right)}\right)^{\prime }dx=\frac{1}{\sqrt{x - 1} \sqrt{x + 1}} dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{1 d x}=x$$$（步驟見 »）。

該積分變為

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

令 $$$u=x^{2} - 1$$$。

則 $$$du=\left(x^{2} - 1\right)^{\prime }dx = 2 x dx$$$ (步驟見»)，並可得 $$$x dx = \frac{du}{2}$$$。

該積分變為

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$：

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

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=- \frac{1}{2}$$$：

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

回顧一下 $$$u=x^{2} - 1$$$：

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

因此，

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

加上積分常數：

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

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