I am going through a proof of the combinatorial result $$\sum\_{k=0}^{n}{n \choose k}(-1)^k \frac{1}{k+1} = \frac{1}{n+1}, $$ and have found something I don't understand.

The proof is as follows:

$$\sum\_{k=0}^{n}{n \choose k}(-1)^k \frac{1}{k+1} = \sum\_{k=0}^{n}{n \choose k}(-1)^k \int\_0^1 t^k\ dt = \int\_0^1 (1-t)^n\ dt = \frac{1}{n+1}.$$

I understand this, except from the second to last equality. **Why does this equality hold?**

Any help would be appreciated!