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calculate-u-nm-0-e-nx-ln-1-e-mx-dx-find-n-0-and-m-0-u-nm-




Question Number 125233 by mathmax by abdo last updated on 09/Dec/20
calculate u_(nm) =∫_0 ^∞ e^(−nx) ln(1+e^(mx) )dx  find Σ_(n≥0 and m≥0)   u_(nm)
$$\mathrm{calculate}\:\mathrm{u}_{\mathrm{nm}} =\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{nx}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{mx}} \right)\mathrm{dx} \\ $$$$\mathrm{find}\:\sum_{\mathrm{n}\geqslant\mathrm{0}\:\mathrm{and}\:\mathrm{m}\geqslant\mathrm{0}} \:\:\mathrm{u}_{\mathrm{nm}} \\ $$

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