How does one show that, $\forall \ n \in \mathbb{N}$ $$\int \frac{\ln^n|\tan(x)|}{\sin(x)}dx = \frac{e^{i\pi n}(\ln|e^{ix}-1|-\ln|e^{ix}+1|)}{2^n}+C$$ Is this possible to do without using any complex analysis theory like the residue theorem or likewise? I'm looking for a single variable solution, but all the methods are welcome!
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2If you are given the answer you can differentiate and show that it equals to the integrand. :) – Dec 17 '13 at 21:03
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@LinearAlgebra Everyone knows that's cheating! ;) – Tim Ratigan Dec 17 '13 at 21:05
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True, but that's not what I'm interested in. It's just algebraically tedious to do so. Exactly, I'm not into cheating either :) – Parseval Dec 17 '13 at 21:06
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Aside from the wise method in the comment, if you did not know the answer, you might write $$ \frac{\log^n|\tan x|}{\sin x} = \frac{1}{\tan x} \log^n \tan x \sec x.$$ Since the derivative of $\tan$ is $\sec^2,$ you are almost there...
Igor Rivin
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Still can't show it. Mind pulling me all the way in to the harbour? This is no homework or anything by the way, I was just playing around with Maple and found this interesting. – Parseval Dec 20 '13 at 20:22