sin與cos的積分方法
2012-05-13 9:40 am
回答 (2)
2012-05-13 11:47 pm
✔ 最佳答案
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2012-05-17 02:35:01 補充：
另一計算方法：
∫x sinθ cosθ dθ
= (x/4) ∫(2 sinθ cosθ) d2θ
= (x/4) ∫sin2θ d2θ
= -(x/4) cos2x
所以 ∫(0→π/2) x sinθ cosθ dθ
= -(x/4) cos2x |(0→π/2)
= [-(x/4) cosπ] - [-(x/4) cos0]
= (x/4) + (x/4)
= x/2 ...... (答案)
參考: miraco, miraco
2012-05-14 8:24 am
miraco 厲害佩服！
另提做法供參考：
cos2θ對θ微分= (-sin2θ)‧2 = -2sin2θ
所以sin2θ對θ積分 = -1/2cos2θ
sinθcosθ = 1/2sin2θ 對θ積分=- 1/4cos2θ
θ從0積到 π/2 = -1/4x ( cos2*π/2-cos2*0 ) = -1/4x (-1-1) = -1/4x (-2) = 1/2x
另提做法供參考：
cos2θ對θ微分= (-sin2θ)‧2 = -2sin2θ
所以sin2θ對θ積分 = -1/2cos2θ
sinθcosθ = 1/2sin2θ 對θ積分=- 1/4cos2θ
θ從0積到 π/2 = -1/4x ( cos2*π/2-cos2*0 ) = -1/4x (-1-1) = -1/4x (-2) = 1/2x
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原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20120513000015KK00766