Let, #I=int(cos2x-cos2alpha)/(cosx-cosalpha)dx#.
Then, #I=int{(2cos^2x-1)-(2cos^2alpha-1)}/(cosx-cosalpha)dx#,
#=2int(cos^2x-cos^2alpha)/(cosx-cosalpha)dx#,
#=2int{(cosx-cosalpha)(cosx+cosalpha)}/(cosx-cosalpha)dx#,
#=2int(cosx+cosalpha)dx=2intcosxdx+2cosalphaint1dx#.
# rArr I=2sinx+2xcosalpha+C#.