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

1 Answer
Konstantinos Michailidis
Feb 24, 2016

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

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