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Logarithmic and Trigonometric Integral

Posted: Sat May 12, 2018 7:36 am
by jacks
$$\int^{\frac{\pi}{6}}_{0}\ln^2(2\sin x)dx$$

Re: Logarithmic and Trigonometric Integral

Posted: Sat May 12, 2018 9:22 am
by Riemann
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 $\dfrac{7 \pi^3}{216}$.

Re: Logarithmic and Trigonometric Integral

Posted: Sat May 12, 2018 11:23 am
by jacks
Thanks Riemann answer is Right. would you like to explain me in detail.