$$$\sqrt{x^{2} - 25}$$$ 的积分

求$$$\int \sqrt{x^{2} - 25}\, dx$$$。

设$$$x=5 \cosh{\left(u \right)}$$$。

则$$$dx=\left(5 \cosh{\left(u \right)}\right)^{\prime }du = 5 \sinh{\left(u \right)} du$$$（步骤见»）。

此外，可得$$$u=\operatorname{acosh}{\left(\frac{x}{5} \right)}$$$。

被积函数变为

$$$\sqrt{x^{2} - 25} = \sqrt{25 \cosh^{2}{\left( u \right)} - 25}$$$

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

$$$\sqrt{25 \cosh^{2}{\left( u \right)} - 25}=5 \sqrt{\cosh^{2}{\left( u \right)} - 1}=5 \sqrt{\sinh^{2}{\left( u \right)}}$$$

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

$$$5 \sqrt{\sinh^{2}{\left( u \right)}} = 5 \sinh{\left( u \right)}$$$

所以，

$${\color{red}{\int{\sqrt{x^{2} - 25} d x}}} = {\color{red}{\int{25 \sinh^{2}{\left(u \right)} d u}}}$$

对 $$$c=25$$$ 和 $$$f{\left(u \right)} = \sinh^{2}{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{25 \sinh^{2}{\left(u \right)} d u}}} = {\color{red}{\left(25 \int{\sinh^{2}{\left(u \right)} d u}\right)}}$$

应用降幂公式 $$$\sinh^{2}{\left(\alpha \right)} = \frac{\cosh{\left(2 \alpha \right)}}{2} - \frac{1}{2}$$$，并令 $$$\alpha= u $$$:

$$25 {\color{red}{\int{\sinh^{2}{\left(u \right)} d u}}} = 25 {\color{red}{\int{\left(\frac{\cosh{\left(2 u \right)}}{2} - \frac{1}{2}\right)d u}}}$$

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

$$25 {\color{red}{\int{\left(\frac{\cosh{\left(2 u \right)}}{2} - \frac{1}{2}\right)d u}}} = 25 {\color{red}{\left(\frac{\int{\left(\cosh{\left(2 u \right)} - 1\right)d u}}{2}\right)}}$$

逐项积分：

$$\frac{25 {\color{red}{\int{\left(\cosh{\left(2 u \right)} - 1\right)d u}}}}{2} = \frac{25 {\color{red}{\left(- \int{1 d u} + \int{\cosh{\left(2 u \right)} d u}\right)}}}{2}$$

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

$$\frac{25 \int{\cosh{\left(2 u \right)} d u}}{2} - \frac{25 {\color{red}{\int{1 d u}}}}{2} = \frac{25 \int{\cosh{\left(2 u \right)} d u}}{2} - \frac{25 {\color{red}{u}}}{2}$$

设$$$v=2 u$$$。

则$$$dv=\left(2 u\right)^{\prime }du = 2 du$$$ (步骤见»)，并有$$$du = \frac{dv}{2}$$$。

因此，

$$- \frac{25 u}{2} + \frac{25 {\color{red}{\int{\cosh{\left(2 u \right)} d u}}}}{2} = - \frac{25 u}{2} + \frac{25 {\color{red}{\int{\frac{\cosh{\left(v \right)}}{2} d v}}}}{2}$$

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

$$- \frac{25 u}{2} + \frac{25 {\color{red}{\int{\frac{\cosh{\left(v \right)}}{2} d v}}}}{2} = - \frac{25 u}{2} + \frac{25 {\color{red}{\left(\frac{\int{\cosh{\left(v \right)} d v}}{2}\right)}}}{2}$$

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

$$- \frac{25 u}{2} + \frac{25 {\color{red}{\int{\cosh{\left(v \right)} d v}}}}{4} = - \frac{25 u}{2} + \frac{25 {\color{red}{\sinh{\left(v \right)}}}}{4}$$

回忆一下 $$$v=2 u$$$:

$$- \frac{25 u}{2} + \frac{25 \sinh{\left({\color{red}{v}} \right)}}{4} = - \frac{25 u}{2} + \frac{25 \sinh{\left({\color{red}{\left(2 u\right)}} \right)}}{4}$$

回忆一下 $$$u=\operatorname{acosh}{\left(\frac{x}{5} \right)}$$$:

$$\frac{25 \sinh{\left(2 {\color{red}{u}} \right)}}{4} - \frac{25 {\color{red}{u}}}{2} = \frac{25 \sinh{\left(2 {\color{red}{\operatorname{acosh}{\left(\frac{x}{5} \right)}}} \right)}}{4} - \frac{25 {\color{red}{\operatorname{acosh}{\left(\frac{x}{5} \right)}}}}{2}$$

因此，

$$\int{\sqrt{x^{2} - 25} d x} = \frac{25 \sinh{\left(2 \operatorname{acosh}{\left(\frac{x}{5} \right)} \right)}}{4} - \frac{25 \operatorname{acosh}{\left(\frac{x}{5} \right)}}{2}$$

使用公式 $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$，化简该表达式：

$$\int{\sqrt{x^{2} - 25} d x} = \frac{5 x \sqrt{\frac{x}{5} - 1} \sqrt{\frac{x}{5} + 1}}{2} - \frac{25 \operatorname{acosh}{\left(\frac{x}{5} \right)}}{2}$$

进一步化简：

$$\int{\sqrt{x^{2} - 25} d x} = \frac{x \sqrt{x - 5} \sqrt{x + 5} - 25 \operatorname{acosh}{\left(\frac{x}{5} \right)}}{2}$$

加上积分常数：

$$\int{\sqrt{x^{2} - 25} d x} = \frac{x \sqrt{x - 5} \sqrt{x + 5} - 25 \operatorname{acosh}{\left(\frac{x}{5} \right)}}{2}+C$$

$$$\int \sqrt{x^{2} - 25}\, dx = \frac{x \sqrt{x - 5} \sqrt{x + 5} - 25 \operatorname{acosh}{\left(\frac{x}{5} \right)}}{2} + C$$$A