Dec 29, 2013 Thread starter #1 S Shobhit Member Nov 12, 2013 23 Show that $$\int_0^{\frac{\pi}{4}}\log\left( \sqrt{\sin^3 x}+\sqrt{\cos^3 x}\right) \text{d}x = \dfrac{G}{12}-\dfrac{5\pi}{16}\log{2}+\dfrac{\pi}{8}\log{\left(3-2\sqrt{2}\right)}+\frac{\pi}{3}\log{\left(2+\sqrt{3} \right)}$$ \(G\) denotes the Catalan's Constant.

Show that $$\int_0^{\frac{\pi}{4}}\log\left( \sqrt{\sin^3 x}+\sqrt{\cos^3 x}\right) \text{d}x = \dfrac{G}{12}-\dfrac{5\pi}{16}\log{2}+\dfrac{\pi}{8}\log{\left(3-2\sqrt{2}\right)}+\frac{\pi}{3}\log{\left(2+\sqrt{3} \right)}$$ \(G\) denotes the Catalan's Constant.