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Question Number 74542 by Cmr 237 last updated on 25/Nov/19

$$$$$$$$$$$$$$\int \left(\mathit{l}\mathit{n}\left(\mathit{x}\right){\mathit{e}}^{\mathit{x}}\right)\mathit{d}\mathit{x}=???$$

Answered by MJS last updated on 25/Nov/19

$$\mathrm{by}\mathrm{parts}$$$$u=\ln x\Rightarrow u^{\prime }=\frac{1}{x}$$$$v^{\prime }=v={\mathrm{e}}^x$$$$\int {\mathrm{e}}^x\ln xdx={\mathrm{e}}^x\ln x-\int \frac{{\mathrm{e}}^x}{x}dx=$$$$={\mathrm{e}}^x\ln x-\mathrm{Ei}x+C$$$$\mathrm{Ei}\left(x\right)\mathrm{is}\mathrm{the}\mathrm{Integral}-\mathrm{Exponential}\mathrm{Function}$$$$\mathrm{please}\mathrm{search}\mathrm{the}\mathrm{web}\mathrm{for}\mathrm{details}$$

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