First, calculate the indefinite integral
#I=int(sin^3x-cosxcotx)dx#
#=intsin^3xdx-intcoscotxdx#
#=I_1-I_2#
#I_1=intsin^3xdx=intsin^2xsinxdx#
#=int(1-cos^2x)sinxdx#
Let #u=cosx#, #=>#, #du=-sinxdx#
#I_1=int-(1-u^2)du#
#=u^3/3-u#
#=1/3cos^3x-cosx#
#I_2=intcoscotxdx#
#=int(cos^2xdx)/(sinx)#
#=int((1-sin^2x)dx)/(sinx)#
#=intcscxdx-intsinxdx#
#=-ln(cscx+cotx)+cosx#
Finally,
#I=1/3cos^3x-2cosx+ln(|cscx+cotx|)+C#
The definite integral is
#int_(1/8pi) ^(11/12pi)(sin^3x-cosxcotx)dx#
#=[1/3cos^3x-2cosx+ln(|cscx+cotx|)]_(1/8pi)^(11/12pi)#
#=(1/3cos^3(11/12pi)-2cos(11/12pi)+ln(|csc(11/12pi)+cot(11/12pi)|))-(1/3cos^3(1/8pi)-2cos(1/8pi)+ln(|csc(1/8pi)+cot(1/8pi)|))#
#=-0.426#
