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I am trying to solve the integral: $$ \int_{0}^{2\pi}d\phi\text{ } \text{ln}(|\cos(\phi)|)^2 $$ I was wondering if the analytical expression of this integral is known. With mathematical tools like Mathematica I can only find the numerical expression which is equal to $8.18649$.

Without the squaring it can be solved: $$ \int_0^{2\pi}d\phi \text{ ln}(|\cos(\phi)|) = -2\pi\ln(2) $$

stacksper
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  • Mathematica evaluates $\int dt \ \ln^2(\cos(t))$, to very long expression. Similarly, she finds $\int\limits_0^{\pi/2} dt \ \ln^2(\cos(t))= (\pi^2 +3 \ln^2(4))\pi/24 $, and your integral is 4 times that – Sal Mar 10 '23 at 14:05
  • This is straightforward if you know about the Beta function. Reduce it to the 0–>pi/2 integral and let u=cos phi – FShrike Mar 10 '23 at 14:06
  • https://math.stackexchange.com/q/58654/1118406 – Sine of the Time Mar 10 '23 at 16:50

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