Suppose we are given the following function.
$$f(x)=\frac{1}{2}[x\sqrt{1-x^2} +\sin^{-1}x]$$. Write down the Taylor series expansion about the origin, up to term involving $x^7$, for the function.
This problem was asked in a mathematics exam where students have to solve 40 questions in 150 minutes.
I know the routine method but it is too lengthy.
Is there a more easy("Think out of the box") like approach to solve it.
Your function is just$$\int_0^x\sqrt{1-t^2}\,\mathrm dt.$$