Integral of $$$\frac{1}{\sqrt{a - x^{2}}}$$$ with respect to $$$x$$$

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

Let $$$x=\sqrt{a} \sin{\left(u \right)}$$$.

Then $$$dx=\left(\sqrt{a} \sin{\left(u \right)}\right)^{\prime }du = \sqrt{a} \cos{\left(u \right)} du$$$ (steps can be seen »).

Also, it follows that $$$u=\operatorname{asin}{\left(\frac{x}{\sqrt{a}} \right)}$$$.

Therefore,

$$$\frac{1}{\sqrt{a - x^{2}}} = \frac{1}{\sqrt{- a \sin^{2}{\left( u \right)} + a}}$$$

Use the identity $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$:

$$$\frac{1}{\sqrt{- a \sin^{2}{\left( u \right)} + a}}=\frac{1}{\sqrt{a} \sqrt{1 - \sin^{2}{\left( u \right)}}}=\frac{1}{\sqrt{a} \sqrt{\cos^{2}{\left( u \right)}}}$$$

Assuming that $$$\cos{\left( u \right)} \ge 0$$$, we obtain the following:

$$$\frac{1}{\sqrt{a} \sqrt{\cos^{2}{\left( u \right)}}} = \frac{1}{\sqrt{a} \cos{\left( u \right)}}$$$

Integral becomes

$${\color{red}{\int{\frac{1}{\sqrt{a - x^{2}}} d x}}} = {\color{red}{\int{1 d u}}}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=1$$$:

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

Recall that $$$u=\operatorname{asin}{\left(\frac{x}{\sqrt{a}} \right)}$$$:

$${\color{red}{u}} = {\color{red}{\operatorname{asin}{\left(\frac{x}{\sqrt{a}} \right)}}}$$

Therefore,

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

Add the constant of integration:

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

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