The linear mass density of a thin rod AB of length L varies from A...

$dm = \lambda . dv$

$dI = dm . {x}^{2}$

$= {\lambda}_{0}\left(1 + \dfrac{x}{L}\right) . {x}^{2}dx$

$I = {\lambda}_{0}{∫}_{0}^{1}\left({x}^{2} + \dfrac{{x}^{3}}{L}\right)dx$

$= {\lambda}_{0}{\left|\dfrac{{x}^{3}}{3} + \dfrac{{x}_{4}}{4L}\right|}_{0}^{L}$

$= {\lambda}_{0}\left[\dfrac{{L}^{3}}{3} + \dfrac{{L}_{3}}{4}\right]$

$I = \dfrac{7{\lambda}_{0}{L}^{3}}{12}$

$m = \mathop{∫}_{0}^{L}\lambda dx = \dfrac{3{\lambda}_{0}L}{2}$

$\dfrac{I}{m} = \dfrac{7{L}^{2}}{12 \times \dfrac{3}{2}} = \dfrac{7{L}^{2}}{18}$

$⇒$ $I = \dfrac{7}{18}{mL}^{2}$