<s>[INST] Check for equality of the integrals I need to show the following identity
$$
\int_a^b\frac{dx}{\sqrt{x-x^2}}=\int_{\tfrac{1-\sqrt{1-a}}{2}}^{\tfrac{1-\sqrt{1-b}}{2}}\frac{dx}{\sqrt{x-x^2}}+\int_{\tfrac{1+\sqrt{1-b}}{2}}^{\tfrac{1+\sqrt{1-a}}{2}}\frac{dx}{\sqrt{x-x^2}}
$$
the sum of the integrals on the right side
$\arcsin \left(-\sqrt{-b+1}\right)-\arcsin \left(-\sqrt{-a+1}\right)+\arcsin \left(\sqrt{-a+1}\right)-\arcsin \left(\sqrt{-b+1}\right)$
and
$$\int_a^b\dfrac{dx}{\sqrt{x-x^2}}=\arcsin \left(2b-1\right)-\arcsin \left(2a-1\right).
$$
Is there any trigonometric identity to show that they are the same? How does this show?
 [/INST] The integrals in the right-hand side sum to
$$
2\arcsin\sqrt{1-a}-2\arcsin\sqrt{1-b}
$$
If you look at https://math.stackexchange.com/a/3211273/62967, you'll see that
$$
\arcsin(2x-1)=2\arcsin\sqrt{x}-\frac{\pi}{2}
$$
For $x=1-a$, we have $2x-1=2-2a-1=1-2a$.
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