Find
$\displaystyle \lim_{R\to \infty }\int ^a_b \frac{x}{1+x^2}$
where a=2R and b=-R
please help me with this, i manage to integrate it, but i don't how to find the limit , the answer in ln(2)
thanks
The question is controversial... pratically the integral ...
$\displaystyle \int_{-\infty}^{+ \infty} \frac{x}{1+x^{2}}\ dx = \lim_{a \rightarrow \infty, b \rightarrow -\infty} \ln \sqrt{\frac{1+a^{2}}{1+b^{2}} $ (1)
... doesn't exist if a and b are mutually independent. The integral...
$\displaystyle \text{PV} \int_{-\infty}^{+ \infty} \frac{x}{1+x^{2}}\ dx = \lim_{a \rightarrow \infty} \ln \sqrt{\frac{1+a^{2}}{1+a^{2}} $ (2)
... where PV means 'principal value' is the (1) with a=b and of course it exists and is zero...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$