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

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

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

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

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

被積函數變為

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

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

$$$\frac{1}{\sqrt{- a^{2} \sin^{2}{\left( u \right)} + a^{2}}}=\frac{1}{\sqrt{1 - \sin^{2}{\left( u \right)}} \left|{a}\right|}=\frac{1}{\sqrt{\cos^{2}{\left( u \right)}} \left|{a}\right|}$$$

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

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

積分變為

$${\color{red}{\int{\frac{1}{\sqrt{a^{2} - x^{2}}} d x}}} = {\color{red}{\int{1 d u}}}$$

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

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

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

$${\color{red}{u}} = {\color{red}{\operatorname{asin}{\left(\frac{x}{\left|{a}\right|} \right)}}}$$

因此，

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

加上積分常數：

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

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