Could you please tell whether this answer is correct or not? If it is wrong, please tell the right answer?

in the 2^nd last step u hve got 

¼ * ( – 3/2 cos(2x+1) + cos(6x+3)/6 )

= – 3/8 cos(2x+1) + cos(6x+3)/24, hence your last step has a mistake of multiplication.

also, another easy way to do this ques can be as follows:

put t= 2x+1

so we have ½ *∫sin^3t dt

now, for ∫sin^3t dt, put cost= u and write ∫sin^3t dt= ∫sin^2t * sint dt= ∫(1 – cos^2t) * sint dt and solve further