<s>[INST] primitives of $f(x)=\frac{1}{2\sqrt{x-x^{2}}}$ I have this function $f:(0,1)\rightarrow \mathbb{R},f(x)=\frac{1}{2\sqrt{x-x^{2}}}$   and I need to find the primitives of $f(x)$
So I calculated $\int \frac{1}{2\sqrt{x-x^{2}}}\,dx$ and I got $\frac{1}{2}\arcsin(2x-1)+C$ but the right answer is $\arcsin\sqrt{x}+C$ .Where's my mistake?How to start?
 [/INST] If $\theta=\arcsin \sqrt{x}$, then $\sin\theta=\sqrt{x}$, so $\sin^2\theta=x$. 
Since $\sin^2\theta=\frac{1-\cos(2\theta)}{2}$, we get
$$\frac{1-\cos(2\theta)}{2}=x$$
We deduce $-\cos(2\theta)=2x-1$. 
Then, $\sin(2\theta-\pi/2)=2x-1$. 
So, (maybe up to a constant) we have $\arcsin(2x-1)=2\theta-\pi/2$.
Which means that $\theta=\frac{1}{2}\arcsin(2x-1)+\pi/4$. So you have that $\arcsin \sqrt{x}$ and $\frac{1}{2}\arcsin(2x-1)$ differ by a constant. So, both results are okay.
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