Alternative ways to evaluate $\int_0^1 x^{2n-1}\ln(1+x)dx =\frac{H_{2n}-H_n}{2n}$ https://ift.tt/eA8V8J

For all, $n\geq 1$, prove that

$$\int_0^1 x^{2n-1}\ln(1+x)\,dx =\frac{H_{2n}-H_n}{2n}\tag1$$ This identity I came across to know from here, YouTube which is proved in elementary way.

Trying to make alternate efforts to prove it, we can observe that $$\ln(1+x)=\ln\left(\frac{1-x^2}{1-x}\right)=\ln(1-x^2)-\ln(1-x)$$On multiplying by $x^{2n-1}$ and integrating from $0$ to $1$ , we have first order partial derivatives of beta function.

How can we prove that identity $(1)$ without the use of derivatives of beta function?

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Naren