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!

The following Python + SymPy script:
from sympy import *

# symbolic variable
x = Symbol('x')

# compute and print Maclaurin series
print series(asin(x),x,0,11,'+')

produces the following output:
x + x**3/6 + 3*x**5/40 + 5*x**7/112 + 35*x**9/1152 + O(x**11)

Therefore, assuming that the implementation is correct, the answer to your question is:
$$\displaystyle\arcsin(x) \approx x + \frac{1}{6} x^3 + \frac{3}{40} x^5 + \frac{5}{112} x^7 + \frac{35}{1152} x^9$$
