1 + cos2x

1 + cos2x

Input
1 + cos(2 x)
Roots
x = pi n – pi/2, n element Z
Series expansion at x=0
2 – 2 x^2 + (2 x^4)/3 + O(x^5)
(Taylor series)
Derivative
d/dx(1 + cos(2 x)) = -2 sin(2 x)
Indefinite integral
integral (1 + cos(2 x)) dx = x + sin(x) cos(x) + constant
Global maxima
max{1 + cos(2 x)} = 2 at x = pi n for integer n
Definite integral over a half-period
integral_0^(pi/2) (1 + cos(2 x)) dx = pi/2~~1.5708
Definite integral over a period
integral_0^pi (1 + cos(2 x)) dx = pi~~3.14159
Definite integral mean square
integral_0^pi (1 + cos(2 x))^2/pi dx = 3/2 = 1.5

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