Example 3. Find $\int \arcsin x d x$.
Solution. Let

$$
u=\arcsin x, d v=d x
$$

we determine

$$
d u=\frac{1}{\sqrt{1-x^{2}}} d x, v=x
$$

Therefore,

$$
\begin{gathered}
\int \arcsin x d x=x \arcsin x-\int x \frac{1}{\sqrt{1-x^{2}}} d x= \\
=x \arcsin x+\frac{1}{2} \int\left(1-x^{2}\right)^{-\frac{1}{2}} d\left(1-x^{2}\right)=x \arcsin x+ \\
+\frac{1}{2} \frac{\left(1-x^{2}\right)^{\frac{1}{2}}}{\frac{1}{2}}+C=x \arcsin x+\sqrt{1-x^{2}}+C .
\end{gathered}
$$