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Logarithmic and Trigonometric Integral

Real & Complex Analysis, Calculus & Multivariate Calculus, Functional Analysis,
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jacks
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Logarithmic and Trigonometric Integral

#1

Post by jacks » Sat May 12, 2018 7:36 am

$$\int^{\frac{\pi}{6}}_{0}\ln^2(2\sin x)dx$$
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Riemann
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Re: Logarithmic and Trigonometric Integral

#2

Post by Riemann » Sat May 12, 2018 9:22 am

A hint is along these lines. Apply the sub $x=\arctan t$ and use the well known fact that

$$\sin \left ( \arctan t \right ) = \frac{t}{\sqrt{t^2+1}}$$

The final answer is $\dfrac{7 \pi^3}{216}$.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
jacks
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Re: Logarithmic and Trigonometric Integral

#3

Post by jacks » Sat May 12, 2018 11:23 am

Thanks Riemann answer is Right. would you like to explain me in detail.
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