Integrate $\int_0^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$ integrate $$\int_0^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$
I've started by dividing this into two integrals:
$$\int_0^{1/2} \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$
and
$$\int_{1/2}^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$
Then I'm trying to find a primitive to 
$$\int \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$ using substitution. However I don't succeed with this. Using the integral from wolframalpha:
https://www.wolframalpha.com/input/?i=integrate+1%2F%28sqrt%28x%281-x%29%29%29
I still fail to find the answer that should be $\pi$.

$$x(1-x)=\frac{-(4x^2-4x)}4=\frac{1-(2x-1)^2}4$$
Set $2x-1=\sin\theta$
$$\int_0^1 \frac{\mathrm{d}x}{\sqrt{x(1-x)}}$$
$$=2\int_{-\frac\pi2}^\frac\pi2\frac{\cos\theta}{\cos\theta}\frac{d\theta}2$$
$$=\frac\pi2-\left(-\frac\pi2\right)$$
