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Question Number 63881 by mmkkmm000m last updated on 10/Jul/19

$$\int e^x/Lnxdx$$

Commented by mathmax by abdo last updated on 11/Jul/19

$$letA=\int \frac{e^x}{lnx}dxchangementlnx=tgive$$$$A=\int \frac{e^{e^t}}{t}{e}^{t}dt=\int \frac{e^{e^t+t}}{t}dtwehave{e}^z=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\Rightarrow$$$$e^{e^t+1}=\sum _{n=0}^{\mathrm{\infty }}\frac{{(e^t+t)}^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}\left(\sum _{k=0}^n{C}_n^k{e}^{kt}{t}^{n-k}\right)\Rightarrow$$$$A=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}(\sum _{k=0}^n{C}_n^k\int e^{kt}{t}^{n-k}dt)$$$$=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}\sum _{k=0}^n{W}_k+Cwih{W}_k=\int e^{kt}{t}^{n-k}dt$$

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