Chapter 6.4, Problem 213E

In the following exercises, find the Maclaurin series of $F(x)={\displaystyle \underset{0}{\overset{x}{\int }}f(t)dt}$by integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.

213. $F(x)={\sin }^{-1}x;$ $f(t)=\frac{1}{\sqrt{1-r^2}}={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\left(\frac{\frac{1}{2}}{k}\right)}\frac{t^{2k}}{k!}$