∫0πcos2xlnsinxdx=

I=∫0πcos2xlnsinxdx=2∫0π2cos2xlnsinxdx                                           =2[sin2x2lnsinx|0π2−∫0π2sin2x2cotxdx]                                           =−2∫0π2cos2xdx=−2⋅12⋅π2=−π2