Show using antiderivatives and using the binomial coefficient.
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$$f(x)=(x-1)^n=\sum_{k=0}^n{n\choose k}(-1)^{n-k}x^k$$ then $$\int_0^x f(x)=\frac{(x-1)^{n+1}-(-1)^{n+1}}{n+1}=\sum_{k=0}^n{n\choose k}(-1)^{n-k}\frac{x^{k+1}}{k+1}$$ now $x=1$.
Nosrati
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how are you getting $\frac{(x-1)^{n+1}\textbf{-1}}{n+1}$? where does the -1 come from? – plantondi Feb 15 '17 at 07:41