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Math Help - improper integrals

  1. #1
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    improper integrals

    Find

    \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
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Rine198 View Post
    Find

    \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: \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?
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  3. #3
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    how do i explain that :

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

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

    thanks
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Rine198 View Post
    how do i explain that :

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

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

    thanks
    The question is controversial... pratically the integral ...

     \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...

     \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

    \chi \sigma
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