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I start with integrals and attempting to figure out these two integrals, but can't move from a spot

  1. $\int x^2 lnx dx$
  2. $\int \frac{lnx}{\sqrt[3]{x}}dx$

The first example - it doesn't look so complicated, but I just can't get the right result. And the second looks pretty complicated, not sure what do to first there.

Thank you

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  • $\begingroup$ For the second one, use the "rationalizing substitution" $ \ u^3 \ = \ x \ $ , then keep in mind that $ \ \ln (u^3) \ = \ 3 \ \ln u \ $ . $\endgroup$ – colormegone May 3 '14 at 18:56
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I will do the first one: Using integration by parts

$$ \int x^2 \ln x dx = \frac{1}{3} \int \ln x d(x^3) = \frac{1}{3} \ln x x^3 - \frac{1}{3} \int x^3 d(\ln x) = \frac{1}{3} \ln x x^3 - \frac{1}{3} \int x^2 dx $$

$$ = \frac{1}{3} \ln x \ x^3 - \frac{1}{9}x^3 + C$$

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  • $\begingroup$ I made a small edit because your text was breaking to the next line at an awkward spot in the equation. $\endgroup$ – colormegone May 3 '14 at 18:55

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