Ta có : $(1+x)^n = C^0_n + C^1_nx+ C^2_nx^2+...+C^n_nx^n, \forall x \in \mathbb{R}$
$\Rightarrow \int\limits_{1}^{2}(1+x)^n dx = \int\limits_{1}^{2}(C^0_n+C^1_nx+C^2_nx^2+...+C^n_nx^n)dx$
$\Rightarrow S=C^0_n + \frac{2^2-1}{2}C^1_n+\frac{2^3-1}{3}C^2_n+...+\frac{2^{n+1}}{n+1}C^n_n=\frac{3^{n+1}-2^{n+1}}{n+1}.$