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I'm making some sort of silly mistake in doing the integral at the final step of showing the arcsine law for discrete random walks. I am supposed to get

$$\int^{xn}_0 \frac{du}{\pi\sqrt{u(n-u)}}=\frac2\pi\sin^{-1}\sqrt x $$

but can't. What I have is

$$\int^{xn}_0 \frac{du}{\pi\sqrt{u(n-u)}}=\\ \frac1\pi\int^{xn}_0 \frac{du}{\sqrt{n^2/4-(u-n/2)^2}}=\\ \frac1\pi\int^{xn-n/2}_{-n/2}\frac{dy}{\sqrt{n^2/4-(y)^2}}=\\ \frac1\pi\sin^{-1}\left.\left(\frac{y}{n/2}\right)\right|^{xn-n/2}_{-n/2}=\\ \frac1\pi\left(\sin^{-1}(2x-1)-\sin^{-1}(-1)\right)=\\ \frac1\pi\sin^{-1}(2x-1)+1/2$$

I'd greatly appreciate being set right.

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