$$\int \ln(\cos x)\,dx$$ In this integral can I substitute $\cos x$ by $e^t$? by proceeding this way i am using byparts but is looping back to the point from which i have started
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1You can try. Then differentiate the answer you come up with to see if it has the derivative you want. – Ethan Bolker Mar 17 '24 at 20:25
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i have integrated it but didn't differentiated that. can you give me insight about can i substitute any function with any other? – Swastik Sanyal Mar 17 '24 at 20:28
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You could also use the complex definition of $\cos$ : $$\cos(\theta)=\frac{1}{2}(e^{i\theta}+e^{-i\theta})$$ – Mar 17 '24 at 20:30
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If you [edit] the question to show us your work we may be able to help you find your error. – Ethan Bolker Mar 17 '24 at 20:33
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HINT: $ \cos x= \mathbb {R} (e^{ix})$ – Narasimham Mar 17 '24 at 20:38
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1You can, but you have to change the differential accordingly, i.e. $$dx=-\dfrac{e^t}{\sqrt{1-e^{2t}}} dt$$ I don't think the correcponding integral will be much easier to solve – Marco Mar 17 '24 at 20:48
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Wolfram Alpha gives an answer involving the polylogarithm function, which seems about right to me. This is not going to have an elementary form. – ConMan Mar 17 '24 at 22:56
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can you please give a introduction to what is polylogarithm function – Swastik Sanyal Mar 18 '24 at 06:24