A rod of length L has non-uniform linear mass density given by  ρx=a+bxL2, where a and b are constants and  0≤x≤L. The value of x for the centre of mass of the rod is at

Given : dmdx=a+bx2L2   ​⇒dm=a+bx2L2dx  Now,  xcm=∫xdm∫dm=∫0Lax+bx3L2dx∫0La+bx2L2dx  ⇒ xcm=ax22+bx44L2ax+bx33L20L​⇒xcm=aL22L+bL44L2aL+bL33L2  ​⇒ xcm=2aL2+bL243aL+bL3    =342a+b3a+bL