Question

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

Answer 1

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

$co{s}^{2}x-si{n}^{2}x=co{s}^{2}x$

$1+cos2x=2co{s}^{2}x$

$\int \frac{cos2x}{\sqrt{1+co{s}^2\left(2x\right)}}dx=\int \frac{co{s}^{2}x}{\sqrt{2co{s}^{2}2x}}dx$

$\int \frac{co{s}^{2}x}{\sqrt{2co{s}^{2}2x}}dx$

$\int \frac{cos2x}{\sqrt{2}cos2x}dx$

$=\int \frac{dx}{\sqrt{2}}=1/\sqrt{2}x$ $=\frac{x}{\sqrt{2}}$ Ans