Chapter 7 - Techniques of Integration - 7.1 Integration by Parts - 7.1 Exercises - Page 517: 51

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

Given $$\int (\ln x)^ndx$$ Let \begin{align*} u&=(\ln x)^n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ dv=dx\\ u&=n(\ln x)^{n-1} \frac{1}{x}dx\ \ \ \ \ \ \ \ \ \ \ v=x \end{align*} Then using integration by parts \begin{align*} \int (\ln x)^ndx&=uv-\int vdu\\ &= x(\ln x)^n -n\int (\ln x)^{n-1} dx \end{align*}