Integral of $$$\frac{1}{\sqrt{1 - x^{2}}}$$$

Find $$$\int \frac{1}{\sqrt{1 - x^{2}}}\, dx$$$.

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

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

Therefore,

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

Add the constant of integration:

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

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