${\mathrm{C}}_0+\frac{{\mathrm{C}}_1\mathrm{x}}{2}+\frac{{\mathrm{C}}_2{\mathrm{x}}^2}{3}+\dots +\frac{{\mathrm{C}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}}{\mathrm{n}+1}=$

$$\begin{gathered} {\int }_0^x{\left(1+t\right)}^{n}dt={\int }_0^x\left(C_0+C_{1}t+C_2{t}^2+.....+C_n{t}^n\right)dt \\ {C}_{0}x+\frac{C_1}{2}{x}^2+\frac{C_2}{3}{x}^3+.....+\frac{C_n}{n+1}{x}^{n+1}=\frac{{\left(1+x\right)}^{n+1}-1}{n+1} \end{gathered}$$

$C_0+\frac{C_1}{2}x+\frac{C_2}{3}{x}^2+.....+\frac{C_n}{n+1}{x}^n=\frac{{\left(1+x\right)}^{n+1}-1}{\left(n+1\right)x}$