# Evaluate $\int_{ - \infty}^{\infty} \frac{dx}{1+x^2}$ using complex integration

I'm trying to evaluate the real integral $$\int_{ - \infty}^{\infty} \frac{dx}{1+x^2}$$

Denote $\Gamma_{1}=\left[-R,R\right]\ \Gamma_{2}=Re^{it}$, for $t\in\left[0,\pi\right]$, and let $\gamma$ be a small circle around $i$ so $\gamma$ is inside the area bounded by $\Gamma_{1}\cup\Gamma_{2}$. By Cauchy's theorem: $$\int_{\Gamma_{1}}f\left(z\right)dz+\int_{\Gamma_{2}}f\left(z\right)dz=\int_{\gamma}f\left(z\right)dz$$ And calculating $\int_{\gamma}f\left(z\right)dz$ gives us $\pi$ (operating Cauchy's formula on the function $\frac{1}{z+i}$). so we got $$\int_{\Gamma_{1}}f\left(z\right)dz+\int_{\Gamma_{2}}f\left(z\right)dz=\pi$$ now I need to show that $$\lim_{R\rightarrow\infty}\int_{\Gamma_{2}}f\left(z\right)dz=0$$ and I'm stuck.

• Search and read about Jordan's lemma. – Taozi May 5 '16 at 7:42

## 1 Answer

You can apply Estimation lemma. Since $$\left|\int_{\Gamma_2} \frac{1}{1+z^2}dz\right| \le \frac{\pi R}{R^2 -1}$$ for large $R$, $$\lim_{R\to\infty}\left|\int_{\Gamma_2} \frac{1}{1+z^2}dz\right|=0.$$ Then you can get what you want.

• As simple as it is beautiful! Thank you! – Uria Mor May 5 '16 at 7:48