Question

$\int \frac{{\cos }^{2}x-{\sin }^{2}x}{\sqrt{1+cos4x}}dx$

Answer 1

$\int \frac{{\cos }^{2}x-{\sin }^{2}x}{\sqrt{1+\cos 4x}}dx$

$=\int \frac{\cos 2x}{\sqrt{1+2{\cos }^{2}2x-1}}dx$

$=\int \frac{\cos 2xdx}{\sqrt{2}\cos 2x}$

$=\frac{1}{\sqrt{2}}\int dx$

$=\frac{x}{\sqrt{2}}$