What is the integration of (ln(x))^n?

[saeed69]

what is the integration of (ln(x))^n?

 

[HallsofIvy]

To integrate ln x by parts, let u= ln x, dv= dx. Then du= (1/x)dx and v= x so
$$\int ln x= x(ln x)- \int dx= xln x- x+ C$$

To integrate $(ln x)^2$ by parts, let u= ln x, dv= ln x dx. Then du= (1/x)dx and v= xln x- x so
$$x(ln x)^2- x ln x- \int ln x- 1 dx= x(ln x)^2- x ln x- (x ln x - x- x)= x(ln x)^2- 2x+ C$$

Keep integrating by parts until you see a pattern.