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

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    Integrating Ln
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SUMMARY

The integration of the function $$\int (\ln x)^p dx$$ where $$p > 0$$ can be approached using the substitution $$u = \ln(x)$$, leading to the integral $$\int u^{p} e^{u} du$$. For integer values of $$p$$, tabular integration by parts is recommended for simplification. However, when $$p$$ is a real number, the result involves the incomplete Gamma function, which complicates the integration process. Understanding these methods is essential for effectively tackling this type of integral.

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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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