## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{(\arcsin x)^{2}+1}{\sqrt{1-x^{2}}} d x
$$
## Solution

$$
\begin{aligned}
& \int \frac{(\arcsin x)^{2}+1}{\sqrt{1-x^{2}}} d x=\int \frac{\arcsin ^{2} x}{\sqrt{1-x^{2}}} d x+\int \frac{d x}{\sqrt{1-x^{2}}}= \\
& =\int(\arcsin x)^{2} d(\arcsin x)+\arcsin x=\frac{1}{3} \cdot(\arcsin x)^{3}+\arcsin x+C
\end{aligned}
$$

Source — "http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} 33-30$ "

Categories: Kuznetsov's Problem Book Integrals Problem 3 | Integrals

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## Problem Kuznetsov Integrals 3-31

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