MHB Integrating Ln $(x)^p$: Step-by-Step Guide

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    Integrating Ln
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To integrate the expression ∫(ln x)^p dx for p > 0, the substitution u = ln(x) is recommended, transforming the integral into ∫u^p e^u du. If p is an integer, integration by parts can be applied repeatedly, with tabular integration suggested for efficiency. However, if p is a real number, the integration becomes more complex and involves the incomplete Gamma function. The discussion highlights the need for further study to fully understand the integration for real p values. Overall, the integration approach varies significantly based on whether p is an integer or a real number.
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$$
\int (\ln x)^pdx, \quad p > 0
$$
How do I go about integrating this?
 
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Is $p$ an integer? If so, then you could do $u=\ln(x)$, which leads to
$$\int u^{p}e^{u}\,du,$$
which succumbs to by-parts as many times as you need. I'd recommend tabular integration in that case. If $p$ can be real, you get something a bit more nasty, with the incomplete Gamma function in there. I don't know how you get that result. I'd have to study a bit.
 
Ackbach said:
Is $p$ an integer? If so, then you could do $u=\ln(x)$, which leads to
$$\int u^{p}e^{u}\,du,$$
which succumbs to by-parts as many times as you need. I'd recommend tabular integration in that case. If $p$ can be real, you get something a bit more nasty, with the incomplete Gamma function in there. I don't know how you get that result. I'd have to study a bit.

p is a real number.
 
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