What's the Maclaurin series for $\arcsin(x)$?
You did a great job:
Just small mistakes.
To find constant of integration, substitute known value of $\arcsin(x)$, $x=0$ is a good choice. You need to have a well defined interval while dealing with inverse trigonometric functions.
The formula to integrate is $$\int x^n \, dx=\frac{x^{n+1}}{n+1}$$
You're integrating the right hand side incorrectly. The integral of $\dfrac{x^2}{2}$ isn't $\dfrac{2x^3}{3}$, it's $\dfrac{x^3}{2\cdot 3} = \dfrac{x^3}{6}.$ :)))