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PostPosted: Sat May 12, 2018 7:36 am 

Joined: Thu Nov 12, 2015 5:26 pm
Posts: 100
Location: Himachal Pradesh (INDIA)
$$\int^{\frac{\pi}{6}}_{0}\ln^2(2\sin x)dx$$


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PostPosted: Sat May 12, 2018 9:22 am 

Joined: Sat Nov 14, 2015 6:32 am
Posts: 139
Location: Melbourne, Australia
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 $\dfrac{7 \pi^3}{216}$.

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$


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PostPosted: Sat May 12, 2018 11:23 am 

Joined: Thu Nov 12, 2015 5:26 pm
Posts: 100
Location: Himachal Pradesh (INDIA)
Thanks Riemann answer is Right. would you like to explain me in detail.


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