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Question Number 164974 by mnjuly1970 last updated on 24/Jan/22

$$$$$${\int }_0^1\frac{\ln (1-x).\ln \left(x\right)}{x^{\frac{3}{2}}}dx\stackrel{?}{=}{\pi }^2-8\ln \left(2\right)$$$$---m.n---$$$$$$

Answered by qaz last updated on 24/Jan/22

$${\int }_0^1\frac{\ln (1-\mathrm{x})\mathrm{lnx}}{{\mathrm{x}}^{3/2}}\mathrm{dx}$$$$=-\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\frac{1}{\mathrm{n}}{\int }_0^1{\mathrm{x}}^{\mathrm{n}-3/2}\mathrm{lnxdx}$$$$=\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\frac{1}{\mathrm{n}{(\mathrm{n}-\frac{1}{2})}^2}$$$$=\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}[\frac{4}{\mathrm{n}}-\frac{4}{\mathrm{n}-\frac{1}{2}}+\frac{2}{{(\mathrm{n}-\frac{1}{2})}^2}]$$$$={4\mathrm{H}}_{-1/2}+8\cdot (1-2^{-2})\cdot \frac{{\pi }^2}{6}$$$$=-8\ln 2+{\pi }^2$$

Commented by mnjuly1970 last updated on 24/Jan/22

$$thanksalotsirqaz$$

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