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badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1Now, I'm pretty sure\[u=\ln x\]\[1=\frac{dx}{du}\frac{1}{x}\]\[dx=x\,du=e^{\ln x}\,du=e^u\,du\]\[\int\frac{e^u}{u}\,du\]And I could swear that this can be expressed as some disgusting power series variant.

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1\[\int\left(\frac{1}{u}e^u\right)\,du=\int\left(\frac{1}{u}\sum_{n=1}^\infty u^n\right)\,du\]

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1@Zarkon Check this out please. :3

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1\[\int\sum_{n=1}^\infty u^{n1}\,du\]

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1Whoops, forgot something!

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.2there is no antiderivative in terms of elementary functions

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1Expressing the answer in terms of a sum would be fine.

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1\[\int\frac{1}{u}e^u\,du=\int\frac{1}{u}\sum_{n=1}^\infty\frac{u^n}{n!}\,du=\int\sum_{n=1}^\infty\frac{u^{n1}}{n!}\,du=\sum_{n=1}^\infty\frac{u^n}{n\cdot n!}\]

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1\[\sum_{n=1}^\infty\frac{u^n}{n\cdot n!}=\sum_{n=1}^\infty\frac{\ln(x)^n}{n\cdot n!}\]

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1Can it be further simplified?

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1A better question: what does\[\sum_{n=1}^\infty\frac{\ln(x)^n}{n\cdot n!}\]converge to?

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1By the ratio test it converges.

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.2you should include your constant of integration along with the radious of convergence.

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0just a correction :\[\int \frac{1}{u} \sum_{n=0}^{\infty} \frac{u^n}{n!} du=\int (\frac{1}{u}+\sum_{n=1}^{\infty} \frac{u^{n1}}{n!})\ du=\ln u+\sum_{n=1}^{\infty} \frac{u^{n}}{n.n!}+C\]
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