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

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=-1$$$ and $$$f{\left(x \right)} = \frac{1}{\sqrt{1 - x^{2}}}$$$:

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

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,

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

Add the constant of integration:

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

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