$$$\sqrt{x^{2} + y^{2}}$$$ 對 $$$x$$$ 的積分

求$$$\int \sqrt{x^{2} + y^{2}}\, dx$$$。

令 $$$x=\sinh{\left(u \right)} \left|{y}\right|$$$。

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

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

因此，

$$$\sqrt{x^{2} + y^{2}} = \sqrt{y^{2} \sinh^{2}{\left( u \right)} + y^{2}}$$$

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

$$$\sqrt{y^{2} \sinh^{2}{\left( u \right)} + y^{2}}=\sqrt{\sinh^{2}{\left( u \right)} + 1} \left|{y}\right|=\sqrt{\cosh^{2}{\left( u \right)}} \left|{y}\right|$$$

$$$\sqrt{\cosh^{2}{\left( u \right)}} \left|{y}\right| = \cosh{\left( u \right)} \left|{y}\right|$$$

所以，

$${\color{red}{\int{\sqrt{x^{2} + y^{2}} d x}}} = {\color{red}{\int{y^{2} \cosh^{2}{\left(u \right)} d u}}}$$

套用降冪公式 $$$\cosh^{2}{\left(\alpha \right)} = \frac{\cosh{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$，令 $$$\alpha= u $$$:

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

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 $$$f{\left(u \right)} = y^{2} \left(\cosh{\left(2 u \right)} + 1\right)$$$：

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

Expand the expression:

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

逐項積分：

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

配合 $$$c=y^{2}$$$，應用常數法則 $$$\int c\, du = c u$$$：

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

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

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

令 $$$v=2 u$$$。

則 $$$dv=\left(2 u\right)^{\prime }du = 2 du$$$ (步驟見»)，並可得 $$$du = \frac{dv}{2}$$$。

因此，

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

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

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

雙曲餘弦的積分為 $$$\int{\cosh{\left(v \right)} d v} = \sinh{\left(v \right)}$$$：

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

回顧一下 $$$v=2 u$$$：

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

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

$$\frac{y^{2} \sinh{\left(2 {\color{red}{u}} \right)}}{4} + \frac{y^{2} {\color{red}{u}}}{2} = \frac{y^{2} \sinh{\left(2 {\color{red}{\operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}} \right)}}{4} + \frac{y^{2} {\color{red}{\operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}}}{2}$$

因此，

$$\int{\sqrt{x^{2} + y^{2}} d x} = \frac{y^{2} \sinh{\left(2 \operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)} \right)}}{4} + \frac{y^{2} \operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}{2}$$

使用公式 $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$，化簡該表達式：

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

進一步化簡：

$$\int{\sqrt{x^{2} + y^{2}} d x} = \frac{x \sqrt{x^{2} + y^{2}} + y^{2} \operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}{2}$$

加上積分常數：

$$\int{\sqrt{x^{2} + y^{2}} d x} = \frac{x \sqrt{x^{2} + y^{2}} + y^{2} \operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}{2}+C$$

$$$\int \sqrt{x^{2} + y^{2}}\, dx = \frac{x \sqrt{x^{2} + y^{2}} + y^{2} \operatorname{asinh}{\left(\frac{x}{\left|{y}\right|} \right)}}{2} + C$$$A