Cos2x cos2x

Cos2x cos2x

Input
cos(2 x) cos(2 x)
Result
cos^2(2 x)
Roots
x = (pi n)/2 – pi/4, n element Z
Series expansion at x=0
1 – 4 x^2 + (16 x^4)/3 + O(x^5)
(Taylor series)
Derivative
d/dx(cos(2 x) cos(2 x)) = -2 sin(4 x)
Indefinite integral
integral cos^2(2 x) dx = 1/8 (4 x + sin(4 x)) + constant
Definite integral over a half-period
integral_0^(pi/4) cos^2(2 x) dx = pi/8~~0.392699
Definite integral over a period
integral_0^(pi/2) cos^2(2 x) dx = pi/4~~0.785398
Definite integral mean square
integral_0^(pi/2) (2 cos^4(2 x))/pi dx = 3/8 = 0.375

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