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I need help to solve the integral $$\int x^n \ln^n(x)dx$$ I've substituted $u=\ln(x)$ and ended up with $\int u^n e^{(n+1)u}du $ is there a way to proceed with the gamma function or create a series by doing by parts? Thank you for your time.

hwood87
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    Note that if you need to evaluate it for a particular value of $n$, you can proceed backward by derivation of a polynomial in $\ln(x)$ (see this similar post https://math.stackexchange.com/a/3128123/399263), this is often faster that carrying on all the IBPs. Of course it's no use if it is for a general $n$, in which case you go for Eevee's proposal. – zwim Sep 01 '21 at 21:22

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$ \newcommand{\I}{\mathcal{I}} \newcommand{\d}{\mathrm{d}} $You can use integration by parts to get a reduction formula. Fix $m \in \Bbb N$. Let

$$\I_n := \int x^m \ln(x)^n \, \d x$$

Then one iteration of integration by parts (differentiating $\ln(x)^n$ and integrating $x^m$) gives us

$$\I_n = \frac{1}{m+1} x^{m+1} \ln(x)^n - \frac{n}{m+1} \int x^m \ln(x)^{n-1} \, \d x$$

i.e.

$$\I_n = \frac{1}{m+1} x^{m+1} \ln(x)^n - \frac{n}{m+1} \I_{n-1}$$

(Of course, we have $m=n$, but you don't want to use this until the very end.) Your preferred means of solving recurrence relations can be used from here.

PrincessEev
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