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In formula 4.224 12 in the 8. edition of the Table of Integrals, Series, and Products is an error: In the case $a^2=1$ both equations should hold, but they are not equal. I think the second equation (i.e. $\int_0^\pi \ln(1+a\cos x)dx = 2 \pi \ln \frac{|a|}{2}$ ($a^2 \geq 1$) is wrong and it should be $$\int_0^\pi \ln((1+a\cos x)^2)dx = 2 \pi \ln \frac{|a|}{2}$$ This would also solve the problem, that the argument of the logarithm gets negative if $a^2 \geq 1$. Can someone confirm this and/or does someone know how to prove this?

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this is entry 172 in the 2005 Errata, which somehow didn't make it properly into the 8th edition.

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