Other ways to evaluate the integral $\int_{-\infty}^{\infty} \frac{1}{1+x^{2}} \, dx$? 
$$\int_{-\infty}^{\infty}\frac{1}{x^2+1}\,dx=\pi $$

I can do it with the substitution $x= \tan u$ or complex analysis. Are there any other ways to evaluate this?

\begin{align}
\int_{-\infty}^{\infty}\frac{dx}{1+x^2}&=2\int_{0}^{\infty}\frac{dx}{1+x^2}\\
&=2\int_{0}^{1}\frac{dx}{1+x^2}+2\int_{1}^{\infty}\frac{dx}{1+x^2}\\
&=4\int_{0}^{1}\frac{dx}{1+x^2}\\
&=4\int_{0}^{1}\sum_{j=0}^{\infty}\left(-x^2\right)^jdx\\
&=4\sum_{j=0}^{\infty}\left(-1\right)^j\int_{0}^{1}\left(x^2\right)^jdx\\
&=4\sum_{j=0}^{\infty}\frac{(-1)^j}{2j+1}\\
&=4\times \frac{\pi}{4}
\end{align}
Note
