chinmay...both ur answers are right..

1st consider..

answer a :log[sin(pi/4 +x)] + c

answer b: log[rt2sin(pi/4+x)] + d {i am using 'd' to indicate tht it is  a diff constant}

answer b can be written as log(rt2)+log[sin(pi/4 +x)] +d as logab=loga+logb

thts all.. the 'c' u got in the 1st case is equal to 'd' + log(rt2)..thts the only diff!!!

the mistake u have made is tht u have considered c,d as the same which is not true in this case