Prove that $\sum_{r=1}^n \frac 1{r}\binom{n}{r} = \sum_{r=1}^n \frac 1{r}(2^r - 1)$

$$\sum_{r=1}^n \frac{1}{r}\binom{n}{r}=\int_{0}^1\sum_{r=1}^n\binom{n}{r}x^{r-1}dx=\int_{0}^1\frac{(1+x)^r-1}{x}dx\\=\int_{0}^1\sum_{r=1}^n (1+x)^{r-1}dx\\=\sum_{r=1}^n \frac{2^r-1}{r}$$

Solution without using calculus:

$$\begin{align} \sum_{r=1}^n\frac 1r\binom nr &=\sum_{r=1}^n\frac 1r\sum_{k=r}^n\binom kr-\binom{k-1}r \qquad\qquad\qquad\text{(*)}\\ &=\sum_{r=1}^n\frac 1r\sum_{k=r}^n\binom {k-1}{r-1}\\ &=\sum_{r=1}^n\sum_{k=r}^n\frac1k\binom kr\\ &=\sum_{k=1}^n\frac 1k\sum_{r=1}^k\binom kr\\ &=\sum_{k=1}^n\frac 1k(2^k-1)\\ &=\sum_{r=1}^n\frac 1r(2^r-1)\qquad\blacksquare \end{align}$$ * by "untelescoping" $\binom nr$ and noting that $\binom {r-1}r=0$.