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we know that $$ \ln(x)=\frac{1}{x} \int_{0}^{1}dt \frac{1}{1+xt} $$

then given a cuadrature formula inside $(0,1)$ is that true

$$ \ln(x)= \frac{1}{x}\sum_{i}\frac{w_{i}}{1+xt_{i}} $$

wht other rational approximations of logarithm are useful based on rational functions ?? $ R(x)$

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We know this? I don't. It appears to me that the integral is incorrect. –  Ron Gordon May 23 '13 at 19:27
    
@RonGordon, I also thought the integral was incorrect, but if we perform the change of variables $u=1+xt$, then $du=xdt$, this $x$ cancels the other one outside, and $u$ ranges from $1$ to $x$ as in the usual definition. –  user1620696 May 23 '13 at 19:31
    
@user1620696: no, $dt=du/x$ and the $x$ outside does not cancel. You end up with $\log{(1+x)}/x^2$. –  Ron Gordon May 23 '13 at 19:33
    
@RonGordon, sorry, I didn't note that, I thought the wrong way arround. It's really incorrect. –  user1620696 May 23 '13 at 19:34
    
so my expression should be $ ln(x)= x\int_{0}^{1} \frac{dt}{1+xt} $ –  Jose Garcia May 23 '13 at 19:37
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