We have
I = int (lnx)^2/x^2dxI=∫(lnx)2x2dx
If z = lnxz=lnx then dz = dx/xdz=dxx and
z = ln xz=lnx so e^z = xez=x
I = int z^2e^(-z)dzI=∫z2e−zdz
Integrating by parts
u = z^2u=z2 so du = 2z dzdu=2zdz
dv = e^(-z) dzdv=e−zdz so v = -e^(-z)v=−e−z
I = u*v - int v duI=u⋅v−∫vdu
I = -z^2e^(-z) + 2intze^(-z)dzI=−z2e−z+2∫ze−zdz
Integrating by parts once again
u = zu=z so du = dzdu=dz
dv = e^(-z)dzdv=e−zdz so v = -e^(-z)v=−e−z
I = -z^2e^(-z) + 2(-ze^(-z) + inte^(-z)dz)I=−z2e−z+2(−ze−z+∫e−zdz)
I = -z^2e^(-z) - 2ze^(-z) - 2e^(-z) + cI=−z2e−z−2ze−z−2e−z+c
I = -e^(-z)(z^2 + 2z + 2) + cI=−e−z(z2+2z+2)+c
But we want to have an answer in terms of xx so if we remember that z = ln(x)z=ln(x) we'll have
I = -e^(-ln(x))((lnx)^2 + 2lnx + 2) +cI=−e−ln(x)((lnx)2+2lnx+2)+c
I = c-((lnx)^2 + 2lnx + 2)/xI=c−(lnx)2+2lnx+2x
