Question

What is the antiderivative of `(ln(x))^2`?

Answer 1

It is #int (lnx)^2dx= x*[(lnx)^2-2*lnx+2]+c#

Explanation:

The antiderivative is using integration by parts

#int (lnx)^2dx=int x'(lnx)^2dx=x*(lnx)^2-int x[(lnx)^2]'dx= x*(lnx)^2-int x*2*lnx*1/xdx= x*(lnx)^2-2*int lnxdx= x*(lnx)^2-2*[x*lnx-int x*1/xdx]= x*(lnx)^2-2*x*lnx+2*x+c= x*[(lnx)^2-2*lnx+2]+c#