$$ (1+\mathrm{x})^{12}={ }^{12} \mathrm{C}_0+{ }^{12} \mathrm{C}_1 \mathrm{x}+{ }^{12} \mathrm{C}_2 \mathrm{x}^2+\ldots \ldots \ldots+{ }^{12} \mathrm{C}_{12} \mathrm{x}^{12} $$ integrating both sides $$ \left(\frac{(1+x)^{13}}{13}\right)_{-2}^2=\left({ }^{12} C_0 x+{ }^{12} C_1 \frac{x^2}{2}+{ }^{12} C_2 \frac{x^3}{3}+\ldots+{ }^{12} C_{12} \frac{x^{13}}{13}\right)_{-2}^2 $$