$$$\frac{1}{x^{2} \sqrt{64 - x^{2}}}$$$ 的積分

求$$$\int \frac{1}{x^{2} \sqrt{64 - x^{2}}}\, dx$$$。

令 $$$x=8 \sin{\left(u \right)}$$$。

則 $$$dx=\left(8 \sin{\left(u \right)}\right)^{\prime }du = 8 \cos{\left(u \right)} du$$$（步驟見»）。

此外，由此可得 $$$u=\operatorname{asin}{\left(\frac{x}{8} \right)}$$$。

被積函數變為

$$$\frac{1}{x^{2} \sqrt{64 - x^{2}}} = \frac{1}{64 \sqrt{64 - 64 \sin^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}}$$$

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

$$$\frac{1}{64 \sqrt{64 - 64 \sin^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}}=\frac{1}{512 \sqrt{1 - \sin^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}}=\frac{1}{512 \sqrt{\cos^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}}$$$

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

$$$\frac{1}{512 \sqrt{\cos^{2}{\left( u \right)}} \sin^{2}{\left( u \right)}} = \frac{1}{512 \sin^{2}{\left( u \right)} \cos{\left( u \right)}}$$$

因此，

$${\color{red}{\int{\frac{1}{x^{2} \sqrt{64 - x^{2}}} d x}}} = {\color{red}{\int{\frac{1}{64 \sin^{2}{\left(u \right)}} d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{64}$$$ 與 $$$f{\left(u \right)} = \frac{1}{\sin^{2}{\left(u \right)}}$$$：

$${\color{red}{\int{\frac{1}{64 \sin^{2}{\left(u \right)}} d u}}} = {\color{red}{\left(\frac{\int{\frac{1}{\sin^{2}{\left(u \right)}} d u}}{64}\right)}}$$

將被積函數改寫為以餘割函數表示:

$$\frac{{\color{red}{\int{\frac{1}{\sin^{2}{\left(u \right)}} d u}}}}{64} = \frac{{\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}}{64}$$

$$$\csc^{2}{\left(u \right)}$$$ 的積分是 $$$\int{\csc^{2}{\left(u \right)} d u} = - \cot{\left(u \right)}$$$：

$$\frac{{\color{red}{\int{\csc^{2}{\left(u \right)} d u}}}}{64} = \frac{{\color{red}{\left(- \cot{\left(u \right)}\right)}}}{64}$$

回顧一下 $$$u=\operatorname{asin}{\left(\frac{x}{8} \right)}$$$：

$$- \frac{\cot{\left({\color{red}{u}} \right)}}{64} = - \frac{\cot{\left({\color{red}{\operatorname{asin}{\left(\frac{x}{8} \right)}}} \right)}}{64}$$

因此，

$$\int{\frac{1}{x^{2} \sqrt{64 - x^{2}}} d x} = - \frac{\sqrt{1 - \frac{x^{2}}{64}}}{8 x}$$

化簡：

$$\int{\frac{1}{x^{2} \sqrt{64 - x^{2}}} d x} = - \frac{\sqrt{64 - x^{2}}}{64 x}$$

加上積分常數：

$$\int{\frac{1}{x^{2} \sqrt{64 - x^{2}}} d x} = - \frac{\sqrt{64 - x^{2}}}{64 x}+C$$

$$$\int \frac{1}{x^{2} \sqrt{64 - x^{2}}}\, dx = - \frac{\sqrt{64 - x^{2}}}{64 x} + C$$$A