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Im having trouble integrating this function.

$$\int_0^{0.5}\int_0^{\pi/4} \frac{r\sin\theta \ln(1-r\cos\theta -r\sin\theta )}{r\cos\theta-\sqrt{(r\cos\theta)^2+(r\sin\theta)^2 }} r \,d\theta \, dr$$

I can simplify this to:

$$\int_0^{0.5}\int_0^{\pi/4} \frac{r\sin\theta \ln(1-r\cos\theta -r\sin\theta )}{r\cos\theta-\sqrt{r^2 }} r \,d\theta \,dr$$

Im stuck on the integration.

Can anyone help me, thanks in advance

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  • $\begingroup$ Keep simplifying, you get $$\int_0^{0.5}\int_0^{\pi/4} \frac{r\sin\theta \ln(1-r\cos\theta -r\sin\theta )}{\cos\theta-1} \,d\theta \,dr$$ $\endgroup$
    – user9464
    May 26, 2016 at 1:22
  • $\begingroup$ @Jack why can that be done? also can the top line be simplified, i cant find anything $\endgroup$
    – someguy
    May 26, 2016 at 1:43

1 Answer 1

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Keep simplifying, we get

$$ I = -\int_{0}^{\frac{1}{2}}\int_{0}^{\frac{\pi}{4}} r \cot\left(\frac{\theta}{2}\right) \log\left(1-r\sqrt{2}\sin\left(\theta+\frac{\pi}{4}\right)\right)\,d\theta\,dr$$ but $\cot\left(\frac{\theta}{2}\right)\sin\left(\theta+\frac{\pi}{4}\right) $ has a non-integrable singularity in a right neighbourhood of $\theta=0$, hence the above integral is simply not converging.

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