1. ## improper integrals

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

2. Originally Posted by Rine198
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
Spoiler:
Hint: $\displaystyle \lim_{R\to\infty}\frac{1}{2}\ln\left(\frac{1+4R^2} {1+R^2}\right)= \frac{1}{2}\ln\left(\lim_{R\to\infty}\frac{1+4R^2} {1+R^2}\right)$

Can you proceed?

3. how do i explain that :

$\displaystyle \int ^a_b \frac{x}{1+x^2 }$

where $\displaystyle a=\infty$ and $\displaystyle b=-\infty$ is divergent?

thanks

4. Originally Posted by Rine198
how do i explain that :

$\displaystyle \int ^a_b \frac{x}{1+x^2 }$

where $\displaystyle a=\infty$ and $\displaystyle b=-\infty$ is divergent?

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

$\displaystyle \chi$ $\displaystyle \sigma$