integral [1 to 2] (lnx)^2/(x^3)

how would i solve this

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- Jul 29th 2010, 07:21 AMkensingtonhelp needed integral [1 to 2] (lnx)^2/(x^3)
integral [1 to 2] (lnx)^2/(x^3)

how would i solve this - Jul 29th 2010, 07:35 AMmathaddict
- Jul 29th 2010, 07:35 AMPlato
- Jul 29th 2010, 07:44 AMHallsofIvy
$\displaystyle \int \frac{(ln(x))^2}{x^3}dx$

When I see a ln(x)**and**an x in the denominator, the first thing I think of is to substitute u= ln(x). Then $\displaystyle du= \frac{1}{x} dx$. In this problem that still leaves $\displaystyle x^2$ in the denominator but if u= ln(x), then $\displaystyle x= e^u$ and $\displaystyle \frac{1}{x^2}= e^{-2u}$.

The integral becomes $\displaystyle \int ue^{-2u}du$ which can be done by integration by parts.