$$$\frac{\sqrt{- a^{2} + x^{2}}}{x}$$$ 對 $$$x$$$ 的積分

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

令 $$$x=\cosh{\left(u \right)} \left|{a}\right|$$$。

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

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

因此，

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

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

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

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

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

積分可以改寫為

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\left|{a}\right|$$$ 與 $$$f{\left(u \right)} = \frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}}$$$：

$${\color{red}{\int{\frac{\sinh^{2}{\left(u \right)} \left|{a}\right|}{\cosh{\left(u \right)}} d u}}} = {\color{red}{\left|{a}\right| \int{\frac{\sinh^{2}{\left(u \right)}}{\cosh{\left(u \right)}} d u}}}$$

將分子與分母同乘一個雙曲餘弦，並使用公式 $$$\cosh^2\left(\alpha \right)=\sinh^2\left(\alpha \right)+1$$$（取 $$$\alpha= u $$$），把其餘部分都以雙曲正弦表示:

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

令 $$$v=\sinh{\left(u \right)}$$$。

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

該積分變為

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

重寫並拆分分式:

$$\left|{a}\right| {\color{red}{\int{\frac{v^{2}}{v^{2} + 1} d v}}} = \left|{a}\right| {\color{red}{\int{\left(1 - \frac{1}{v^{2} + 1}\right)d v}}}$$

逐項積分：

$$\left|{a}\right| {\color{red}{\int{\left(1 - \frac{1}{v^{2} + 1}\right)d v}}} = \left|{a}\right| {\color{red}{\left(\int{1 d v} - \int{\frac{1}{v^{2} + 1} d v}\right)}}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dv = c v$$$：

$$\left|{a}\right| \left(- \int{\frac{1}{v^{2} + 1} d v} + {\color{red}{\int{1 d v}}}\right) = \left|{a}\right| \left(- \int{\frac{1}{v^{2} + 1} d v} + {\color{red}{v}}\right)$$

$$$\frac{1}{v^{2} + 1}$$$ 的積分是 $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$：

$$\left|{a}\right| \left(v - {\color{red}{\int{\frac{1}{v^{2} + 1} d v}}}\right) = \left|{a}\right| \left(v - {\color{red}{\operatorname{atan}{\left(v \right)}}}\right)$$

回顧一下 $$$v=\sinh{\left(u \right)}$$$：

$$\left|{a}\right| \left(- \operatorname{atan}{\left({\color{red}{v}} \right)} + {\color{red}{v}}\right) = \left|{a}\right| \left(- \operatorname{atan}{\left({\color{red}{\sinh{\left(u \right)}}} \right)} + {\color{red}{\sinh{\left(u \right)}}}\right)$$

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

$$\left|{a}\right| \left(\sinh{\left({\color{red}{u}} \right)} - \operatorname{atan}{\left(\sinh{\left({\color{red}{u}} \right)} \right)}\right) = \left|{a}\right| \left(\sinh{\left({\color{red}{\operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}}} \right)} - \operatorname{atan}{\left(\sinh{\left({\color{red}{\operatorname{acosh}{\left(\frac{x}{\left|{a}\right|} \right)}}} \right)} \right)}\right)$$

因此，

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

加上積分常數：

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

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