d
\(\mathrm{x}_{\mathrm{cm}}=\frac{\int \mathrm{xdm}}{\int \mathrm{dm}}=\frac{\int(\lambda \mathrm{dx}) \mathrm{x}}{\int \mathrm{dm}}\)

\(=\frac{\int_{0}^{L}\left(a+\frac{b x^{2}}{L^{2}}\right) x d x}{\int_{0}^{L}\left(a+\frac{b x^{2}}{L^{2}}\right) d x}\)

\(=\frac{\frac{a L^{2}}{2}+\frac{b}{L^{2}} \cdot \frac{L^{4}}{4}}{a L+\frac{b}{L^{2}} \cdot \frac{L^{3}}{3}}\)

\(=\frac{\left(\frac{4 a+2 b}{8}\right) L}{\frac{(3 a+b)}{3}}=\frac{3}{4} \frac{(2 a+b) L}{(3 a+b)}\)