200. $\int \frac{d x}{\sqrt{a^{2}-x^{2}}}$.

200. $\int \frac{d x}{\sqrt{a^{2}-x^{2}}}$. 

(Note: The original text and the translation are identical as the text was already in English.)
Solution. Let's bring this integral to the standard form IX:

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

Using the substitution $x=a t ; d x=a d t$. Therefore,

$$
\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\frac{1}{a} \int \frac{a d t}{\sqrt{1-t^{2}}}=\int \frac{d t}{\sqrt{1-t^{2}}}=\arcsin t+C=\arcsin \frac{x}{a}+C
$$

Thus,

$$
\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\arcsin \frac{x}{a}+C
$$

It is useful to remember formulas (3) and (4) and use them as standard integrals.