2cosx-1

2cosx-1

Input
2 cos(x) – 1
Alternate form
e^(-i x) + e^(i x) – 1
Series expansion at x=0
1 – x^2 + x^4/12 + O(x^6)
(Taylor series)
Derivative
d/dx(2 cos(x) – 1) = -2 sin(x)
Indefinite integral
integral (-1 + 2 cos(x)) dx = 2 sin(x) – x + constant
Global maxima
max{2 cos(x) – 1} = 1 at x = 2 pi n for integer n
Definite integral over a half-period
integral_0^pi (-1 + 2 cos(x)) dx = -pi~~-3.14159
Definite integral over a period
integral_0^(2 pi) (-1 + 2 cos(x)) dx = -2 pi~~-6.28319
Definite integral mean square
integral_0^(2 pi) (-1 + 2 cos(x))^2/(2 pi) dx = 3

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