ln sinx 在π/6到5π/6的定积分

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ln sinx 在π/6到5π/6的定积分

ln sinx 在π/6到5π/6的定积分
ln sinx 在π/6到5π/6的定积分

ln sinx 在π/6到5π/6的定积分
∫[π/6,5π/6]lnsinxdx
=∫[π/6,π/2]lnsinxdx+∫[π/2,5π/6]lnsinxdx
sinx=cos(π/2-x)
=∫[π/6,π/2)lncos(π/2-x)dx+∫[π/2,5π/6]lncos(π/2-x)dx
π/2-x=u x=π/2-u
= -∫[π/3,0]lncosudu -∫[0,-π/3]lncosudu
=∫[0,π/3]lncosudu-∫[0,-π/3]lncosudu
=∫[0,π/3]lncosudu+∫[-π/3,0]lncosudu
=∫[-π/3,π/3]lncosudu