Compute the following real integral using the residue theorem I'm trying to compute the following integral using the residue theorem: $$\int_{-\infty}^\infty \frac{dx}{x^2-2x+4}$$
I know that this integral is equal to the integral over the upper half plane since the integral over the semi-circle in the upper half plane goes to zero as $R \to \infty$. I'm happy with computing this integral and have found the result to be $\pi/\sqrt{3}$ (which I'm fairly confident is correct). 
However, I'm having trouble showing that the integral over the semi-circle in the upper half plane does go to zero. I know that I can use the $Ml$ inequality to show this. 
Let the radius of the semi-circle be $R$. So I know that $l$ is $2\pi R$. So then I need to find an upper bound on the modulus of the integrand. This is where I'm stuck. How do I do this?
 A: Complex plane
In the complex plane we use the function
$$
f(z) = \frac{1}{z^{2}-2z+4}
$$

Find poles
Where does $z^{2}-2z+4=0$? When
$$
 z = 1 \pm i\sqrt{3}
$$
These poles are represented as $\color{red}{\times}$ in the figure below.
Contour

Jordan lemma
$$
\oint f(z)\, dz = \lim_{R\to\infty} \left(
\int_{\Gamma_{R}} f(z)\, dz + 
\int_{\Omega_{R}} f(z)\, dz
\right)
$$
Therefore,
$$
\int_{-\infty}^{\infty} f(x)\, dx = 
\oint f(z)\, dz 
+ \lim_{R\to\infty} \left(
\int_{\Gamma_{R}} f(z)\, dz 
\right)
$$
Because 
$$
 \lim_{z\to\infty} |z\, f(z)| = \lim_{z\to\infty} \Bigg|\frac{z}{z^{2}-2z+4}\Bigg| = 0
$$
$$
\lim_{R\to\infty} \left(
\int_{\Gamma_{R}} f(z)\, dz 
\right) = 0
$$
Residue theorem
$$
 \oint_{\Gamma_{R}} f(z)\, dz = 2\pi i \sum_{k} \text{Res }f(z_{k})
$$
where the points $z_{k}$ are the poles enclosed by the contour. In this instance, there is a single pole at 
$$z_{1} = 1+i\sqrt{3}$$
The Laurent expansion about this point is
$$
  f\left(z-z_{1}\right) = 
-\frac{i}{2 \sqrt{3}}  \frac{1}{\zeta -1 - \sqrt{3}} +
\frac{1}{12} +
\frac{i}{24 \sqrt{3}} \left(\zeta -1-i \sqrt{3}\right) +
\mathcal{O}\left(\zeta -1-i \sqrt{3} \right)^{2}
$$
$$
 \oint_{\Gamma_{R}} f(z)\, dz = 2\pi i \left( -\frac{i}{2 \sqrt{3}} \right) = \frac{\pi}{\sqrt{3}}
$$
Conclusion
$$
\int_{\infty}^{\infty} \frac{dx}{ x^{2}-2x+4 } = \frac{\pi}{\sqrt{3}}
$$
A: You maximal value is on the boundary of your semidisk. Take R o infinity and you are ready. $$\frac{1}{2\pi }\left|\int_{\gamma}\frac{1}{z^2-2z+4}\right|\le \frac{2\pi R}{2\pi}\max_{|z|=R}\left|\frac{1}{z^2-2z+4}\right|\le\frac{R}{R^2-2R+4}\le 1/R \to 0 $$ as $R\to \infty$
