Differentiate (1+x)^n = sum C(n,r)$x^r$: n(1+x)^{n-1} = sum rC(n,r)$x^{r-1}$. Put x = 1: n2^{n-1} = sum rC(n,r). This gives the sum of rC(n,r).

Multiply by x first, then differentiate: x*(1+x)^n = sum C(n,r)*$x^{r+1}$. Differentiating: (1+x)^n + nx(1+x)^{n-1} = sum (r+1)C(n,r)$x^r$.

For sum $r^2$*C(n,r): use the identity $r^2$ = r(r-1) + r. Then sum r(r-1)C(n,r) = n(n-1)2^{n-2}, and sum rC(n,r) = n2^{n-1}. Total: n(n-1)2^{n-2} + n2^{n-1} = n(n+1)*2^{n-2}.

Integrate (1+x)^n from 0 to x: (1+x)^{n+1}/(n+1) - $\frac{1}{n+1}$ = sum C(n,r)*$x^{r+1}$/(r+1). Put x = 1: $\frac{2^{n+1} - 1}{(n+1)}$ = sum C$\frac{n,r}{(r+1)}$.