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Question Number 129275 by abdurehime last updated on 14/Jan/21

$$\int {\mathrm{e}}^{{\mathrm{x}}^{\mathrm{x}}}\mathrm{dx}??????$$

Commented by abdurehime last updated on 14/Jan/21

$$\mathrm{no}\mathrm{one}????$$

Commented by Dwaipayan Shikari last updated on 14/Jan/21

$$\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{n!}\int {\left(x^x\right)}^{n}dx=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{n!}\int {\left(x^x\right)}^n$$$$=\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{n!}\int e^{nxlogx}dx$$$$=\underset{n=1}{\sum ^{\mathrm{\infty }}}\underset{k=1}{\sum ^{\mathrm{\infty }}}\frac{1}{n!}.\frac{n^k}{k!}\int {\left(xlogx\right)}^{k}dx=\sum _{n⩾1}\sum _{k⩾1}\frac{n^k}{n!k!}\int t^k{e}^{(k+1)t}dtlogx=t$$$$=\sum _{n⩾1}\sum _{k⩾1}\frac{n^k}{n!k!}\sum _{m⩾1}\int \frac{t^k{(k+1)}^m{t}^m}{m!}dt$$$$=\sum _{n⩾1}\sum _{k⩾1}\sum _{m⩾1}\frac{n^k{(k+1)}^m{\left(logx\right)}^{k+m+1}}{n!k!m!(k+m+1)}+C$$

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