# 1 + cos2x

Input |
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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 |