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$$I=(-1)^{n-1}{m\over n+m}\int_{0}^{1}x^{n-1}(1-x)^{m-1}dx=\sum_{i=0}^{n}(-1)^i{n\choose i}H_{n+m-i}\tag1$$ Where $H_n$ is the nth harmonic number
Recall
$$H_n=\int_{0}^{1}{1-x^n\over 1-x}dx\tag2$$
Sub $(2)$ into $(1)\rightarrow (3)$
$$I=\sum_{i=0}^{n}(-1)^i{n\choose i}\int_{0}^{1}{1-x^{n+m-i}\over 1-x}dx\tag3$$
Let $$J=\int_{0}^{1}{1-x^{n+m-i}\over 1-x}dx\tag4$$
$$J=\sum_{k=0}^{\infty}\int_{0}^{1}{x^k-x^{k+n+m-i}}dx\tag5$$
$$J=\sum_{k=0}^{\infty}\left({1\over k+1}-{1\over k+1+n+m-i}\right)\tag6$$
Can someone help me here to prove I, don't seem to have a clue where I am going?