Integrating around a dog bone contour Prove that
$$\int^{1}_{0} \sqrt{x}\sqrt{1-x}\,dx = \frac{\pi}{8}$$
$$\textit{proof}$$
Consider the function
$$f(z) = \sqrt{z-z^2} = e^{\frac{1}{2}\log(z-z^2)}$$
Consider the branch cut on the x-axis
$$x(1-x)\geq 0\,\, \implies \, 0\leq x \leq 1 $$
Consider $ w= z-z^2 $ then
$$\log(w) = \log|w|+i\theta,\,\, \theta\in[0,2\pi)$$
Consider the contour

Consider the integral
$$\int_{c_0}f(z)\,dz+\int_{c_1}f(z)\,dz+\int^{1-\epsilon}_{\epsilon} e^{\frac{1}{2}\log|x-x^2|}\,dx-\int^{1-\epsilon}_{\epsilon} e^{\frac{1}{2}\log|x-x^2| +\pi i}\,dx = 2\pi i \mathrm{Res}(f,\infty)$$
Consider the Laurent expansion of
$$\sqrt{z-z^2} =i \sqrt{z^2} \sqrt{1-\frac{1}{z}}= iz\sum_{k=0}^\infty{\frac{1}{2} \choose k} \left(-\frac{1}{z} \right)^k$$
Hence we deuce that
$$ \mathrm{Res}(f,\infty) = -\frac{i}{8}$$
That implies
$$\int_{c_0}f(z)\,dz+\int_{c_1}f(z)\,dz+2\int^{1-\epsilon}_{\epsilon} \sqrt{x}\sqrt{1-x}\,dx = \frac{\pi}{4}$$
Considering integrals the contours around $c_0$ and $c_1$ go to zero . Finally we get
$$\int^{1}_{0} \sqrt{x}\sqrt{1-x}\,dx = \frac{\pi}{8}$$

Question
I have my concerns about the expansion at infinity 
$$\sqrt{z-z^2} =i \sqrt{z^2} \sqrt{1-\frac{1}{z}}= iz\sum_{k=0}^\infty{\frac{1}{2} \choose k} \left(-\frac{1}{z} \right)^k$$
First I am assuming that $\sqrt{z^2} = z$ which seems to be wrong on the chosen branch cut. Also it is wrong to assume that $\sqrt{zw} = \sqrt{z}\sqrt{w}$.
 A: We    may    use    the     method    from    the    following    MSE
link     for    the
residue at infinity. The  introduction and the continuity argument can
be copied verbatim and will not be repeated here. We use
$$f(z) = \exp(1/2\times \mathrm{LogA}(z))
\exp(1/2\times \mathrm{LogB}(1-z))$$
with the  two logarithms defined as  in the linked-to  post. Note that
this is not the choice  Mathematica or Maple makes, but consistency is
sufficient here and we can re-use a vetted computation.
The difference is that we have $f(z)\sim  z$ at infinity so we need to
determine the  coefficients on $z,$  the constant coefficient  and the
one on $1/z.$ There was less work at the linked-to computation because
the function was $\sim 1/z$ at infinity. 
We use $f(z)/z^2$ for the first  one and put $z = R\exp(i\theta).$ Now
the modulus of $\mathrm{LogA}(z)$ is  $\log R.$ We get for the modulus
of $\mathrm{LogB}(1-z)$
$$\log\sqrt{(1-R\cos(\theta))^2 + R^2\sin(\theta)^2}
\\ = \log\sqrt{1-2R\cos(\theta) + R^2}.$$
We are manipulating logarithms of positive real numbers using the real
logarithm and may continue with
$$\log R + \log\sqrt{1-2\cos(\theta)/R+1/R^2}.$$
Using the  leading term  and the method  from the link  we immediately
obtain  $f(z)   \sim  -iz.$  For  the  constant   coefficient  we  use
$(f(z)+iz)/z.$ We have
$$\exp\left(\frac{1}{2}\log\sqrt{1-2\cos(\theta)/R+1/R^2}\right)
= \sqrt[4]{1-2\cos(\theta)/R+1/R^2}
\\ = 1 - \frac{1}{4} (2\cos(\theta)/R-1/R^2)
- \frac{3}{32} (2\cos(\theta)/R-1/R^2)^2 -\cdots
\\ = 1 - \frac{1}{2}\cos(\theta)\frac{1}{R}
+ \left(1/4 - \frac{3}{8}\cos(\theta)^2\right)\frac{1}{R^2}
- \cdots$$
We also have
$$\arctan\left(\frac{-R\sin(\theta)}{1-R\cos(\theta)}\right)
= \arctan\left(\frac{-\sin(\theta)}{1/R-\cos(\theta)}\right)
\\ = \theta + \sin(\theta)\frac{1}{R} + 
\sin(\theta)\cos(\theta)\frac{1}{R^2} + \cdots$$
so that  (this actually holds  everywhere as long as  $\theta$ matches
the range of $\arg\mathrm{LogB}$)
$$\exp\left(\frac{1}{2}i\arg\mathrm{LogB(1-z)}\right)
\\ = \exp\left(\frac{1}{2}i\theta\right)
\left(1 + \frac{1}{2}i\sin(\theta)\frac{1}{R} + 
\frac{1}{2}i\sin(\theta)\cos(\theta)\frac{1}{R^2} -
\frac{1}{8} \sin(\theta)^2\frac{1}{R^2} + \cdots\right).$$
Adding in $iz$ cancels the contribution  from the first terms of these
two  expansions.  We  get  from  the second  term  and collecting  the
contributions from the lower and upper half plane
$$\int_0^{2\pi} \frac{1}{R\exp(i\theta)}
\exp(\log R) \exp(i\theta+\pi i/2)
\\ \times \left(\frac{1}{2}i\sin(\theta)\frac{1}{R}
-\frac{1}{2}\cos(\theta)\frac{1}{R}\right)
Ri\exp(i\theta)
\; d\theta
\\ = \int_0^{2\pi}
\exp(\log R) \exp(i\theta+\pi i/2) i
\left(-\frac{1}{2}\exp(-i\theta)\right) \frac{1}{R}
\; d\theta
\\ = \frac{1}{2} \int_0^{2\pi}
\exp(i\theta)\exp(-i\theta)
\; d\theta = \pi.$$
for a residue of $-(\pi)/(2\pi i)  = i/2.$ This establishes $f(z) \sim
-iz + i/2.$ We  integrate $f(z) + iz - i/2$ to  get the coefficient on
$1/z$, obtaining
$$\int_0^{2\pi}
\exp(\log R) \exp(i\theta+\pi i/2)
\\ \times \frac{1}{R^2} \left(\frac{1}{4}-\frac{3}{8}\cos(\theta)^2
+ \frac{1}{2} i\sin(\theta)\cos(\theta)
- \frac{1}{8} \sin(\theta)^2
-\frac{1}{4}i\sin(\theta)\cos(\theta)\right)
\\ \times Ri\exp(i\theta)
\; d\theta$$
The inner term is
$$\frac{1}{8} - \frac{1}{4}\cos(\theta)^2 + \frac{1}{8}i\sin(2\theta)
= - \frac{1}{8}\cos(2\theta) + \frac{1}{8}i\sin(2\theta)
= - \frac{1}{8}\exp(-2i\theta).$$
which leaves for the integral
$$\frac{1}{8} \int_0^{2\pi} \exp(2i\theta)\exp(-2i\theta) \; d\theta
= \frac{\pi}{4}.$$
We get the residue $-(\pi/4)/(2\pi i) = i/8.$
We have established that at infinity,
$$\bbox[5px,border:2px solid #00A000]{ f(z) \sim
-iz + \frac{1}{2} i + \frac{1}{8} i \frac{1}{z}}$$
and hence $\mathrm{Res}_{z=\infty} f(z) = \frac{1}{8} i.$ Taking  into
account  that the  contour used  with this  branch produces  twice the
value of the integral we obtain
$$\frac{1}{2} \times -2\pi i \times \frac{1}{8} i$$
which is
$$\bbox[5px,border:2px solid #00A000]{ \frac{\pi}{8}.}$$
A: There is a conceptual error in the OP.
Note that if $w=z(1-z)$, then the condition $\arg(w)\in[0,2\pi)$ restricts the domain of the complex $z$ plane to a half space.
To see this, we write $z=|z|e^{i\arg(z)}$ and $1-z=|1-z|e^{i\arg(1-z)}$ so that $w=|z||1-z|e^{i\left(\arg(z)+\arg(1-z)\right)}$.
Inasmuch as $\arg(z)+\arg(1-z)$ spans a range of $4\pi$ in the complex $z$-plane, then $\arg(w)$ does likewise.

To address the concerns in the OP in detail, we begin with a short primer.

PRIMER:

The complex logarithm $\log(z)$ is defined for $z\ne 0$ as
$$\log(z)=\log(|z|)+i\arg(z) \tag 1$$
It is easy to show that the complex logarithm satisfies
$$\log(z_1z_2)=\log(z_1)+\log(z_2) \tag 2$$
which means that any value of $\log(z_1z_2)$ can be expressed as the sum of some value of $\log(z_1)$ and some value of $\log(z_2)$.
To see that $(2)$ is true, we simply note that $\log(|z_1||z_2|)=\log(|z_1|)+\log(z_2)$ and $\arg(z_1z_2)=\arg(z_1)+\arg(z_2)$.


NOTE:  The relationship in $(2)$ is not generally satisfied when the logarithm is restricted on, say, the principal branch for which $\arg(z)=\text{Arg}(z)$, where $-\pi<\text{Arg}\le \pi$.


Using $(2)$, we can write for $z\ne0$, $z\ne 1$
$$\begin{align}
f(z)&=\sqrt{z(1-z)}\\\\
&=e^{\frac12 \log(z(1-z))}\\\\
&=e^{\frac12 \left(\log(z)+\log(1-z)\right)}\\\\
&=e^{\frac12\log(z)}e^{\frac12\log(1-z)}\\\\
&=\sqrt{z}\sqrt{1-z}
\end{align}$$


SELECTING A BRANCH OF $\displaystyle \sqrt{z(1-z)}$

To obtain a specific branch of $\sqrt{z(1-z)}$, we can use a branch of $\sqrt{z}$ and another branch of $\sqrt{1-z}$.
If we select the branches for $\sqrt{z}$ and $\sqrt{1-z}$ to be such that $-\pi<\arg(z)\le \pi$ and $0<\arg(1-z)\le 2\pi$, then the branch of $\sqrt{z(1-z)}$ is such that
$$\sqrt{z(1-z)}=\sqrt{|z||1-z|}e^{i\frac12 (\arg(z)+\arg(1-z))}$$
with $-\pi<\arg(z)+\arg(1-z)\le 3\pi$.
With this choice, it is straightforward to show that $\sqrt{z(1-z)}$ is analytic on $\mathbb{C}\setminus [0,1]$.


EVALUATING THE INTEGRAL

Then, we can write
$$\begin{align}
\oint_C \sqrt{z(1-z)}\,dz&=\int_0^1 \sqrt{x(1-x)}e^{i(0+2\pi)/2}\,dx+\int_1^0\sqrt{x(1-x)}e^{i(0+0)}\,dx\\\\
&=-2\int_0^1 \sqrt{x(1-x)}\,dx \tag 3
\end{align}$$


NOTE:  We bypassed consideration of the contributions to the integral from the circular deformations around the branch points since their contributions vansish in the limit as the radii go to zero.


Using Cauchy's Integral Theorem, the value of the integral $\oint_C \sqrt{z(1-z)}\,dz$ is unaltered by deforming $C$ into a circular contour, centered at the origin, of radius, $R>1$.  Hence, exploiting the analyticity of $\sqrt{z(1-z)}$ for $R>1$, we have
$$\begin{align}
\oint_C \sqrt{z(1-z)}\,dz&=\oint_{R>1}\sqrt{z(1-z)}\,dz\\\\
&=\int_{-\pi}^{\pi}\sqrt{Re^{i\phi}(1-Re^{i\phi})}\,iRe^{i\phi}\,d\phi\\\\
&=-\int_{-\pi}^{\pi} \left(iRe^{i\phi}\right)^2 \left(1-\frac{1}{Re^{i\phi}}\right)^{1/2}\,d\phi\\\\
&=-\int_{-\pi}^{\pi} \left(iRe^{i\phi}\right)^2 \left(1-\frac{1/2}{Re^{i\phi}}-\frac{1/8}{(Re^{i\phi})^2}-\frac{1/16}{(Re^{i\phi})^3}+O\left(\frac1{(Re^{i\phi})^4}\right)\right)\,d\phi\\\\
&\to -\frac{\pi}{4}\,\,\text{as}\,\,R\to \infty \tag 4
\end{align}$$
Finally, putting together $(3)$ and $(4)$ yields
$$\int_0^1 \sqrt{x(1-x)}\,dx=\frac{\pi}{8}$$
as expected.


NOTE:  The expansion leading to $(4)$ is correct given the chosen branches of $\sqrt{z}$ and $\sqrt{1-z}$.  Then,


$$\begin{align}
\sqrt{z(1-z)}&=\sqrt{-z^2\left(1-\frac1z\right)}\\\\
&=e^{\frac12\log(-z^2)+\frac12\log\left(1-\frac1z\right)}\\\\
&=iz \sqrt{1-\frac1z}
\end{align}$$


where we used $\log(-1)=i\pi$ and $\log(z^2)=2\log(z)$.  Then, upon expanding $\sqrt{1-\frac1z}$ in its Laurent series in the annulus $1<z<\infty$, and setting $z=Re^{i\phi}$, we obtain the expansion used to arrive at $(4)$.

A: 1) When you have $\sqrt{z^2}$, technically we have $(e^{i\pi}z)^2$ by the branch cut, but obviously that cancels out to $z.$
2) Also the given assumption is valid here because of the expansion that has been chosen; the "circle" of radius of convergence for z in $\sqrt{1-\frac{1}{z}}$ contains $\infty$ (since the area of convergence is really outside the circle here).
