# Math Help - Improper Integration

1. ## Improper Integration

$\displaystyle\int_{-\infty}^{\infty}\frac{x}{\pi(1+x^2)} \ dx=\left[\frac{\ln|1+x^2|}{2\pi}\right]_{-\infty}^{\infty}$

How do I evaluate this?

2. See this.

3. Originally Posted by dwsmith
$\displaystyle\int_{-\infty}^{\infty}\frac{x}{\pi(1+x^2)} \ dx=\left[\frac{\ln|1+x^2|}{2\pi}\right]_{-\infty}^{\infty}$

How do I evaluate this?
The question is a little controversial, so that it is better, if You want the 'Cauchy principal value' of integral, to specify that as follows...

$\displaystyle \text{PV} \int_{-\infty}^{+\infty} \frac{x}{\pi\ (1+x^{2})}\ dx = \lim_{\xi \rightarrow \infty} \int_{-\xi}^{\+\xi} \frac{x}{\pi\ (1+x^{2})}\ dx =0$

Without the prefix 'PV' the integral...

$\displaystyle \int_{-\infty}^{+\infty} \frac{x}{\pi\ (1+x^{2})}\ dx$

... has to be considered divergent...

Kind regards

$\chi$ $\sigma$