complex analysis - Choosing a contour for this integral

$$\int_{0}^\infty \frac{x^{\frac{1}{2}}}{x^2+1} \ dx$$
by taking the branch cut of $x^{\frac{1}{2}}$ along the positive real axis.

I wasn't sure what contour to choose, so I chose this keyhole contour:

Where the upper line segment is $C_1$ and the lower line segment is $C_2$, with the circular arc of radius $R$ being $C_R$ and the smaller circular arc of radius $r$ be $C_r$.

Can someone please show me how to use my contour to evaluate this?
This is my attempt

---

Note that $$\int_{C} \frac{z^{\frac{1}{2}}}{z^2+1} \ dz = 2\pi i \mathrm{Res}(f,i) + 2\pi i \mathrm{Res}(f,-i)$$

where $C = C_1 + C_2 + C_R + C_r$.
First consider
$$\left|\int_{C_R} \frac{z^{\frac{1}{2}}}{z^2+1}\right| \ dz \leq ML$$
where $M = \mathrm{max}\left\{\left|\frac{z^{\frac{1}{2}}}{z^2+1}\right|:z\in C_R\right\} = \frac{R^{\frac{1}{2}}}{R^2 - 1}$
and $L = \mathrm{length}{C_R} = 2\pi R$.
So that expression goes to zero as $R\rightarrow \infty$.
Similarly,
$$\left|\int_{C_r} \frac{z^{\frac{1}{2}}}{z^2+1}\right| \rightarrow 0$$
as $r\rightarrow 0$.

So it seems that all that's left is to parametrize the remaining segments... but I can't seem to have an expression which has upper limits going to infinity.

Answer

You have by the definition of your brach cut $$(x+ir)^{\frac12}\approx\sqrt{x}+\frac{ir}{2\sqrt{x}}$$ and $$(x-ir)^{\frac12}\approx-\sqrt{x}+\frac{ir}{2\sqrt{x}}$$ for $x>r$. Thus the integrals over the segments from $r+ir$ to $R+ir$ and from $R-ir$ to $r-ir$ approximate twice the searched for integral value.

---

For a more boring region substitute $x=e^u$ to get the transformed integral
$$\int_{-\infty}^\infty \frac{e^{\frac32 u}}{1+e^{2u}}du$$
Now integrate along the contour of the box $[-R,R]+i[0,\pi]$. The integration along the upper side has the integrand for $w=u+i\pi$ so that
$$f(w)=\frac{-i\,e^{\frac32 u}}{1+e^{2u}}$$
plus orientation reversal,
the left and right sides are of size $e^{-R/2}$, so that the limit $R\to\infty$ of the contour integral is $(1+i)$ times the required value, and the only pole is at $w=i\frac\pi2$.