# How to integrate cos^4x

Indefinite integral |
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integral cos^4(x) dx = 1/32 (12 x + 8 sin(2 x) + sin(4 x)) + constant |

Expanded form of the integral |

(3 x)/8 + 1/4 sin(2 x) + 1/32 sin(4 x) + constant |

Series expansion of the integral at x=0 |

x – (2 x^3)/3 + x^5/3 + O(x^6) (Taylor series) |

Definite integral over a half-period |

integral_0^pi cos^4(x) dx = (3 pi)/8~~1.1781 |

Definite integral over a period |

integral_0^(2 pi) cos^4(x) dx = (3 pi)/4~~2.35619 |

Definite integral mean square |

integral_0^(2 pi) (cos^8(x))/(2 pi) dx = 35/128~~0.273438 |