Maclaurin expansion of $\arcsin x$ I'm trying to find the first five terms of the Maclaurin expansion of $\arcsin x$, possibly using the fact that
$$\arcsin x = \int_0^x \frac{dt}{(1-t^2)^{1/2}}.$$
I can only see that I can interchange differentiation and integration but not sure how to go about this. Thanks!

Considering the binomial expansion
\begin{align}
\frac 1{\sqrt{1-u}}&=1+\frac12 u+\frac12\cdot\frac32\frac{u^2}{2!}+\frac12\cdot\frac32\cdot\frac52\frac{u^3}{3!}++\frac12\cdot\frac32\cdot\frac52\cdot\frac 72\frac{u^4}{4!}+\dotsm \\
&=1+\frac12 u+\frac{1\cdot3}{2^2\,2!}u^2 +\frac{1\cdot3\cdot 5}{2^3\,3!}u^3 + +\frac{1\cdot3\cdot 5\cdot 7}{2^4\,4!}u^4\dotsm ,
\end{align}
and substituting $u=x^2$, then integrating term by term, you obtain
\begin{align}
\arcsin x&= x+ \frac12 \frac{x^3}3+\frac{1\cdot3}{2^2\,2!}\frac{x^5}5 +\frac{1\cdot3\cdot 5}{2^3\,3!}\frac{x^7}7 + +\frac{1\cdot3\cdot 5\cdot 7}{2^4\,4!}\frac{x^9}9+\dotsm \\[1ex]
&=x+ \frac12 \frac{x^3}3+\frac{1\cdot3}{2\cdot 4}\frac{x^5}5 +\frac{1\cdot3\cdot 5}{2\cdot4\cdot 6}\frac{x^7}7 +\frac{1\cdot3\cdot 5\cdot 7}{2\cdot 4\cdot 6\cdot 8}\frac{x^9}9+\dotsm \\[1ex]
&=\sum_{n=1}^\infty\frac{(2n-1)!!}{(2n)!!}\frac{x^{2n+1}}{2n+1}.
\end{align}
