$$$- \frac{1}{\sqrt{1 - x^{2}}}$$$ 的积分

求$$$\int \left(- \frac{1}{\sqrt{1 - x^{2}}}\right)\, dx$$$。

对 $$$c=-1$$$ 和 $$$f{\left(x \right)} = \frac{1}{\sqrt{1 - x^{2}}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

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

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

因此，

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

加上积分常数：

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

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