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badreferencesBest ResponseYou've already chosen the best response.1
Now, 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.
 2 years ago

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

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

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

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

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

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

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

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

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

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

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

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

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