Evaluate the following integrals : `int(cos^2x-sin^2x)/(sqrt(1+cos4x))dx`

Explanation

We know that `cos 2x=2cos^2 x-1=cos 2x-sin^2 x`

So, numerator becomes `cos2x – sin2x = cos2x`

Denominator becomes, `4x = 2 × 2x ⇒ 1 + cos4x = 1 + cos(2 xx 2x) ⇒ 1 + cos4x = 2cos^(2)2x`

`I=int(frac{cos 2x}{sqrt(2 cos^2 2x)})`

`=int(frac{cos 2x}{sqrt(2 )cos 2 x})`

`=int (frac{1}{sqrt 2})dx`

`=int(frac{1}{sqrt 2})dx`

`=frac{x}{sqrt 2}+C`