How to find $\int \ln^nx\space dx$

Question

How does one evaluate a function in the form of $$\int \ln^nx\space dx$$ My trusty friend Wolfram Alpha is blabbering about $\Gamma$ functions and I am having trouble following. Is there a method for indefinitely integrating such and expression? Or if there isn't a method how would you tackle the problem?

Answer 1

Let $$F_n=\int \log^n(x) dx$$ so by integration by parts (we derivate $\log^n(x)$) we have $$F_n=x\log^n(x)-n\int\log^{n-1}(x)dx=x\log^n(x)-nF_{n-1}$$ so we find $F_n$ by induction by the relation:

$$\left\{\begin{array}\\ F_0=x+C\\ F_{n}=x\log^n(x)-nF_{n-1},\quad n\geq 1 \end{array}\right.$$

Added$\ $ We can write a simple procedure with Maple which gives the expression of $F_n$ for every $n$ as follow:

We can prove by induction that $$F_n=x\log^n(x)+\sum_{k=1}^{n-1}(-1)^{n-k}\frac{n!}{k!} x\log^k(x)+(-1)^nx+C$$

Answer 2

If $\ln^nx$ denotes $\log(x)^n$, then my hint is to try the simple substitution $e^y = x$, giving $$\int\log(x)^ndx = \int y^ne^ydy$$

Answer 3

Write $x=e^y$, and $\text{d}x=e^y\text{d}y$, and integrate by parts a few times $$\int \ln^n(x)\text{d}x\text=\int y^{n}e^{y}\text{d}y=y^ne^y-n\int y^{n-1}e^y\text{d}y.$$