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Log-Sine Integral

Shobhit

Member
Nov 12, 2013
23
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$$\int_0^{\frac{\pi}{4}}\log\left( \sqrt{\sin^3 x}+\sqrt{\cos^3 x}\right) \text{d}x = \dfrac{G}{12}-\dfrac{5\pi}{16}\log{2}+\dfrac{\pi}{8}\log{\left(3-2\sqrt{2}\right)}+\frac{\pi​}{3}\log{\left(2+\sqrt{3} \right)}$$

\(G\) denotes the Catalan's Constant.