# Cos^2 2x

Input |
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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(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 |