$$$\sqrt{- x^{2} y^{2} + 4}$$$ 對 $$$x$$$ 的積分

求$$$\int \sqrt{- x^{2} y^{2} + 4}\, dx$$$。

令 $$$x=\frac{2 \sin{\left(u \right)}}{\left|{y}\right|}$$$。

則 $$$dx=\left(\frac{2 \sin{\left(u \right)}}{\left|{y}\right|}\right)^{\prime }du = \frac{2 \cos{\left(u \right)}}{\left|{y}\right|} du$$$（步驟見»）。

此外，由此可得 $$$u=\operatorname{asin}{\left(\frac{x \left|{y}\right|}{2} \right)}$$$。

所以，

$$$\sqrt{- x^{2} y^{2} + 4} = \sqrt{4 - 4 \sin^{2}{\left( u \right)}}$$$

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

$$$\sqrt{4 - 4 \sin^{2}{\left( u \right)}}=2 \sqrt{1 - \sin^{2}{\left( u \right)}}=2 \sqrt{\cos^{2}{\left( u \right)}}$$$

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

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

因此，

$${\color{red}{\int{\sqrt{- x^{2} y^{2} + 4} d x}}} = {\color{red}{\int{\frac{4 \cos^{2}{\left(u \right)}}{\left|{y}\right|} d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{4}{\left|{y}\right|}$$$ 與 $$$f{\left(u \right)} = \cos^{2}{\left(u \right)}$$$：

$${\color{red}{\int{\frac{4 \cos^{2}{\left(u \right)}}{\left|{y}\right|} d u}}} = {\color{red}{\left(\frac{4 \int{\cos^{2}{\left(u \right)} d u}}{\left|{y}\right|}\right)}}$$

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

$$\frac{4 {\color{red}{\int{\cos^{2}{\left(u \right)} d u}}}}{\left|{y}\right|} = \frac{4 {\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{\left|{y}\right|}$$

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 $$$f{\left(u \right)} = \cos{\left(2 u \right)} + 1$$$：

$$\frac{4 {\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\right)d u}}}}{\left|{y}\right|} = \frac{4 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 u \right)} + 1\right)d u}}{2}\right)}}}{\left|{y}\right|}$$

逐項積分：

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

配合 $$$c=1$$$，應用常數法則 $$$\int c\, du = c u$$$：

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

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，

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

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 $$$f{\left(v \right)} = \cos{\left(v \right)}$$$：

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

餘弦函數的積分為 $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$：

$$\frac{2 \left(u + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{2}\right)}{\left|{y}\right|} = \frac{2 \left(u + \frac{{\color{red}{\sin{\left(v \right)}}}}{2}\right)}{\left|{y}\right|}$$

回顧一下 $$$v=2 u$$$：

$$\frac{2 \left(u + \frac{\sin{\left({\color{red}{v}} \right)}}{2}\right)}{\left|{y}\right|} = \frac{2 \left(u + \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{2}\right)}{\left|{y}\right|}$$

回顧一下 $$$u=\operatorname{asin}{\left(\frac{x \left|{y}\right|}{2} \right)}$$$：

$$\frac{2 \left(\frac{\sin{\left(2 {\color{red}{u}} \right)}}{2} + {\color{red}{u}}\right)}{\left|{y}\right|} = \frac{2 \left(\frac{\sin{\left(2 {\color{red}{\operatorname{asin}{\left(\frac{x \left|{y}\right|}{2} \right)}}} \right)}}{2} + {\color{red}{\operatorname{asin}{\left(\frac{x \left|{y}\right|}{2} \right)}}}\right)}{\left|{y}\right|}$$

因此，

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

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：

$$\int{\sqrt{- x^{2} y^{2} + 4} d x} = \frac{2 \left(\frac{x \sqrt{- \frac{x^{2} \left|{y}\right|^{2}}{4} + 1} \left|{y}\right|}{2} + \operatorname{asin}{\left(\frac{x \left|{y}\right|}{2} \right)}\right)}{\left|{y}\right|}$$

進一步化簡：

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

加上積分常數：

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

$$$\int \sqrt{- x^{2} y^{2} + 4}\, dx = \left(\frac{x \sqrt{- x^{2} y^{2} + 4}}{2} + \frac{2 \operatorname{asin}{\left(\frac{x \left|{y}\right|}{2} \right)}}{\left|{y}\right|}\right) + C$$$A