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$$\int\frac{(\ln x)^{10}}{x}\,dx$$ All I know is that I am supposed to substitute $u=\ln x$. But can someone please explain to me how to find the anti derivative of $(\ln x)^{10}$. I think we are supposed to use integration by parts on integrating $\ln x$, but we haven't been taught that yet.

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    $\begingroup$ If you take $u = \ln(x)$, what do you get for $du$? $\endgroup$ – Mike Pierce Jan 28 '15 at 3:46
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Hint: Let $u = \ln(x)$, then $\frac{du}{dx} = \frac{1}{x} \implies du = \frac{dx}{x}$

If this is true, what do you see for $$\int \left(\ln(x)\right)^{10}\left(\frac{dx}{x}\right)$$?

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  • $\begingroup$ I'm confused.How did du suddenly equal dx/x? $\endgroup$ – Elsa Jan 28 '15 at 4:28
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    $\begingroup$ I just rearranged some terms. $\frac{\ln(x)^{10}}{x} dx = \ln(x)^{10}\frac{dx}{x}$. I can show more work if you like $\endgroup$ – jameselmore Jan 28 '15 at 4:31
  • $\begingroup$ oh thank you! let me try it out! $\endgroup$ – Elsa Jan 28 '15 at 4:32
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    $\begingroup$ Oh I see, this is a common identity, $\frac{d}{dx}\ln(x) = \frac1x$ $\endgroup$ – jameselmore Jan 28 '15 at 4:33
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    $\begingroup$ wait so the derivative of lnx...OH SHOOT HAHAHA i KEPT DOING THE ANTIDERIVATIVE! Thanks for reminding me hahahaha! now let me try $\endgroup$ – Elsa Jan 28 '15 at 4:34

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