$${I = \int_{x = -L/2}^{x = +L/2} x^2 \left( \frac{M}{L} \right) dx \\ = \frac{M}{3L} [x^3]_{-L/2}^{+L/2} = \frac{M}{3L} \left[ \left( \frac{L}{2} \right)^3 - \left( - \frac{L}{2} \right)^3 \right] \\ = \underline{\underline{\frac{1}{12} ML^2}}}$$