求反常积分∫ln(sinx)dx,上限是π/2,下限是0,希望给出过程,多谢
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![求反常积分∫ln(sinx)dx,上限是π/2,下限是0,希望给出过程,多谢](/uploads/image/z/5939657-17-7.jpg?t=%E6%B1%82%E5%8F%8D%E5%B8%B8%E7%A7%AF%E5%88%86%E2%88%ABln%28sinx%29dx%2C%E4%B8%8A%E9%99%90%E6%98%AF%CF%80%2F2%2C%E4%B8%8B%E9%99%90%E6%98%AF0%2C%E5%B8%8C%E6%9C%9B%E7%BB%99%E5%87%BA%E8%BF%87%E7%A8%8B%2C%E5%A4%9A%E8%B0%A2)
求反常积分∫ln(sinx)dx,上限是π/2,下限是0,希望给出过程,多谢
求反常积分∫ln(sinx)dx,上限是π/2,下限是0,希望给出过程,多谢
求反常积分∫ln(sinx)dx,上限是π/2,下限是0,希望给出过程,多谢
∫ln(sinx)dx上限是π/2,下限是0,将x都改成π/2-x;得
∫ln(sinx)dx上限是π/2,下限是0 = -∫ln(cosx)dx上限是0,下限是π/2
= ∫ln(cosx)dx上限是π/2,下限是0;(*)
同理(* )式中再将x都变成x-π/2,得:
∫ln(sinx)dx上限是π/2,下限是0 =∫ln(sinx)dx上限是π,下限是 π/2;
于是∫ln(sinx)dx上限是π/2,下限是0 =二分之一倍的 ∫ln(sinx)∫ln(sinx)dx上限是π/2,下限是0
∫ln(sinx)dx上限是π/2,下限是0 + ∫ln(cosx)dx上限是π/2,下限是0 +∫ln(2)dx上限是π/2,下限是0
=∫ln(2*sinx*cosx)dx上限是π/2,下限是0
=∫ln(sin2x)dx上限是π/2,下限是0
=(1/2)*∫ln(sinx)dx上限是π,下限是0
=∫ln(sinx)dx上限是π/2,下限是0
所以可得:
∫ln(sinx)dx上限是π/2,下限是0=- ∫ln(2)dx上限是π/2,下限是0
=-π*ln(2)/2
I=∫(0,π/2) ln(sinx)dx =∫(0,π/4) ln(sinx)dx+∫(π/4,π/2) ln(sinx)dx =∫(0,π/4) ln(sinx)dx+∫(π/4,0) ln(sin(π/2-x))d(π/2-x)=∫(0,π/4) ln(sinx)dx+∫(0,π/4) ln(cosx)dx=∫(0,π/4) ln(1/2sin2x)dx=∫(0,π/4) ln(sin2x)dx+πln(1/2)/4=1/2∫(0,π/2) ln(sinx)dx-πln2/4=I/2-πln2/4即I=I/2-πln2/4,所以I=-πln2/2