$$$\frac{1}{2 \sqrt{1 - x^{2}}}$$$ 的积分
您的输入
求$$$\int \frac{1}{2 \sqrt{1 - x^{2}}}\, dx$$$。
解答
对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = \frac{1}{\sqrt{1 - x^{2}}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{\frac{1}{2 \sqrt{1 - x^{2}}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{\sqrt{1 - x^{2}}} d x}}{2}\right)}}$$
$$$\frac{1}{\sqrt{1 - x^{2}}}$$$ 的积分为 $$$\int{\frac{1}{\sqrt{1 - x^{2}}} d x} = \operatorname{asin}{\left(x \right)}$$$:
$$\frac{{\color{red}{\int{\frac{1}{\sqrt{1 - x^{2}}} d x}}}}{2} = \frac{{\color{red}{\operatorname{asin}{\left(x \right)}}}}{2}$$
因此，
$$\int{\frac{1}{2 \sqrt{1 - x^{2}}} d x} = \frac{\operatorname{asin}{\left(x \right)}}{2}$$
加上积分常数：
$$\int{\frac{1}{2 \sqrt{1 - x^{2}}} d x} = \frac{\operatorname{asin}{\left(x \right)}}{2}+C$$
答案
$$$\int \frac{1}{2 \sqrt{1 - x^{2}}}\, dx = \frac{\operatorname{asin}{\left(x \right)}}{2} + C$$$A