#f(theta)=thetasin(-theta)+2cot((7theta)/8)#
#int_(pi/4)^((5pi)/6)f(theta)d theta=int_(pi/4)^((5pi)/6)(thetasin(-theta)+2cot((7theta)/8))d theta#
Applying sum rule
#int_(pi/4)^((5pi)/6)(thetasin(-theta)+2cot((7theta)/8))d theta=int_(pi/4)^((5pi)/6)thetasin(-theta)d theta+int_(pi/4)^((5pi)/6)2cot((7theta)/8)d theta#
#int_(pi/4)^((5pi)/6)thetasin(-theta)d theta=int_(pi/4)^((5pi)/6)theta(-sinthetad theta)#
Integrating by parts
#u=theta#
#(du)/d theta=1#
#du=d theta#
#dv=-sinthetad theta#
#intdv=int(-sintheta)d theta#
#v=costheta#
#intudv=uv-intvdu#
Substituting
#inttheta(-sinthetad theta)=thetacostheta-intcosthetad theta#
#inttheta(-sinthetad theta)=thetacostheta-sintheta#
#int_(pi/4)^((5pi)/6)thetasin(-theta)d theta={thetacostheta-sintheta}_(pi/4)^((5pi)/6)#
#=(5pi)/6cos((5pi)/6)-(pi/4)cos(pi/4)-(sin((5pi)/6)-sin(pi/4))#
#=(5pi)/6xx(-sqrt3/2)-pi/4xx1/sqrt2-(1/2-1/sqrt2)#
Rearranging
#int_(pi/4)^((5pi)/6)thetasin(-theta)d theta=(1/sqrt2-1/2)-(5sqrt3/12+1/(4sqrt2)pi)#
#int_(pi/4)^((5pi)/6)2cot((7theta)/8)d theta={(2(logsin((7theta)/8)))/((7theta)/8)}_(pi/4)^((5pi)/6)#
#=16/7xx{1/((5pi)/6)logsin(7xx(5pi)/6)-1/(pi/4)logsin(7xx(pi/4)}#
#=16/(7pi)xx{6/5xxlogsin((35pi)/6)-4xxlogsin((7pi)/4)}#
#=16/(7pi)xx{logsin^(6/5)((35pi)/6)-logsin^4((7pi)/4)}#
#=16/7xxlog{sin^(6/5)((35pi)/6)/sin^4((7pi)/4)}#
#int_(pi/4)^((5pi)/6)2cot((7theta)/8)d theta=16/7xxlog{sin^(6/5)((35pi)/6)/sin^4((7pi)/4)}#
#int_(pi/4)^((5pi)/6)thetasin(-theta)d theta+int_(pi/4)^((5pi)/6)2cot((7theta)/8)d theta=(1/sqrt2-1/2)-(5sqrt3/12+1/(4sqrt2)pi)+16/7xxlog{sin^(6/5)((35pi)/6)/sin^4((7pi)/4)}#
#int_(pi/4)^((5pi)/6)f(theta)d theta=int_(pi/4)^((5pi)/6)(thetasin(-theta)+2cot((7theta)/8))d theta=(1/sqrt2-1/2)-(5sqrt3/12+1/(4sqrt2)pi)+16/7xxlog{sin^(6/5)((35pi)/6)/sin^4((7pi)/4)}#
