Question

Prove that ${\int }_0^{\frac{32\pi }{3}}\sqrt{1+\cos 2x}dx=22\sqrt{2}-\sqrt{\frac{3}{2}}$

Answer 1

$I_1={\int }_{10\pi }^{32\pi /2}|\cos x|dx$

$={\int }_0^{2\pi /3}|\cos x\setminus dx$

$=2-\frac{\sqrt{3}}{2}$

$I=20\sqrt{2}+\sqrt{2}|2-\frac{\sqrt{3}}{2}$

$=20\sqrt{2}+2\sqrt{2}-\frac{\sqrt{3}\sqrt{2}}{2}$

$=22\sqrt{2}-\sqrt{\frac{3}{2}}$