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March 26, 2017

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what is the anti-derivative of cos^5x

  • math - ,

    cos^5θ = cos^4θ cosθ dθ
    = (1 - sin^2θ)^2 cosθ dθ
    = 1 - 2sin^2θ + sin^4θ cosθ dθ

    Now, note that if u = sinθ then du = cosθ dθ and we have

    1 - 2u^2 + u^4 du

    Easy peasie lemon-squeezy:

    u - 2/3 u^3 + 1/5 u^5 + C
    sinθ - 2/3 sin^3θ + 1/5 sin^5θ + C

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