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I am suppose to find $\int(\ln x)^3$ by using the proof (I have to prove this part first) of

$\int (\ln x)^n dx = x(\ln x)^n - n \int (\ln x)^{n-1} dx$

I can not prove it and I do not know how to work with n powers like that, to me it doesn't even look right.

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    (Joke, sort of): If one is asked to prove an explicit integration formula, one way is to differentiate the "answer."2012-06-08

2 Answers 2

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We have

$\mathrm I(n)=\int (\log x)^ndx$

integrate by parts with

$dx=du$ $\log^n x = v$

$x=u$ $n (\log x)^{n-1} \frac{1}{x}dx=dv$

Spoiler ahead

$\mathrm I(n)=x\log^n x-\int n (\log x)^{n-1}\dfrac{1}{x}x dx$


$\mathrm I(n)=x\log^n x-n\int (\log x)^{n-1} dx$


$\mathrm I(n)=x\log^n x-n \mathrm I (n-1)$

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    When a student says "Do you mean log, or natural log?", I say "yes".2012-06-08
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Integration by parts is a useful technique. You can use $u=(\ln x)^n$ and $v'=1$ in the formula $\int uv' dx = uv-\int vu' dx$.