if Bernstein polynomials defined by

$\displaystyle B_{k,n}(x)=\left(\begin{array}{c}n\\k\end{array}\r ight)x^k (1-x)^{n-k}$

how I can integrate it

this is my work

$\displaystyle \int_{a}^{b} \left(\begin{array}{c}n\\k\end{array}\right) x^k (1-x)^{n-k}dx$

$\displaystyle \int_{a}^{b} \left(\begin{array}{c}n\\k\end{array}\right) x^k \sum_{i=0}^{n-k} \left(\begin{array}{c}n-k\\i\end{array}\right) (-1)^i x^i dx$

since the summation is the Taylor series of $\displaystyle (1-x)^{n-k}$

$\displaystyle \sum_{i=0}^{n-k}\left(\begin{array}{c}n\\k\end{array}\right) \left(\begin{array}{c}n-k\\i\end{array}\right) (-1)^i \int_{a}^{b} x^{k+i} dx$

$\displaystyle \sum_{i=0}^{n-k}\left(\begin{array}{c}n\\k\end{array}\right) \left(\begin{array}{c}n-k\\i\end{array}\right) (-1)^i \left(\frac{x^{k+i+1}}{k+i+1}\mid_a^b\right)$

my work is correct but I think I can't reach the true answer if I continue

the answer is $\displaystyle \frac{1}{n+1}\sum_{i=k+1}^{n+1} B_{i,n+1}(x)$

Thanks very much .