$$$\frac{4}{\sqrt{1 - x^{2}}}$$$ 的積分

求$$$\int \frac{4}{\sqrt{1 - x^{2}}}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=4$$$ 與 $$$f{\left(x \right)} = \frac{1}{\sqrt{1 - x^{2}}}$$$：

$${\color{red}{\int{\frac{4}{\sqrt{1 - x^{2}}} d x}}} = {\color{red}{\left(4 \int{\frac{1}{\sqrt{1 - x^{2}}} d x}\right)}}$$

$$$\frac{1}{\sqrt{1 - x^{2}}}$$$ 的積分是 $$$\int{\frac{1}{\sqrt{1 - x^{2}}} d x} = \operatorname{asin}{\left(x \right)}$$$：

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

因此，

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

加上積分常數：

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

$$$\int \frac{4}{\sqrt{1 - x^{2}}}\, dx = 4 \operatorname{asin}{\left(x \right)} + C$$$A