Let
y=(lnx)^2 /(x^3)
intydx=int(lnx)^2 /(x^3)dx
Let
t=lnx
e^t=x
e^2t=x^2
t^2=(lnx)^2
(dt)/dx=1/x
dt=1/xdx
intydx=int(lnx)^2 /(x^2)(1/xdx)
=intt^2/e^(2t)dt
intydx=intt^2e^(-2t)dt
integrating by parts
intudv=uv-intvdu
u=t^2
du=2tdt
dv=e^-2tdt
v=-1/2e^(-2t)
intt^2e^(-2t)dt=t^2(-1/2e^(-2t))-int(-1/2e^(-2t))(2tdt)
=-t^2/2e^(-2t)+intte^(-2t)dt
=-t^2/2e^(-2t)+I_1
where
I_1=intte^(-2t)dt
integrating by parts
intudv=uv-intvdu
u=t
du=dt
dv=e^(-2t)dt
v=-1/2e^(-2t)
intte^(-2t)dt=t(-1/2e^(-2t))-int(-1/2e^(-2t))dt
=-1/2te^(-2t)+1/2(-1/2)e^(-2t)
intte^(-2t)dt=-1/2te^(-2t)-1/4e^(-2t)
I_1=-1/2te^(-2t)-1/4e^(-2t)
intt^2e^(-2t)dt=-t^2/2e^(-2t)+I_1
intt^2e^(-2t)dt=-t^2/2e^(-2t)+(-1/2te^(-2t)-1/4e^(-2t))
intt^2e^(-2t)dt=-t^2/2e^(-2t)-1/2te^(-2t)-1/4e^(-2t)
intt^2e^(-2t)dt=-1/4(2t^2+2t+1)e^-2t
Replacing
t=lnx
e^(-2t)=1/x^2
int(lnx)^2/x^2(1/x)dx=-1/4(2(lnx)^2+2lnx+1)(1/x^2)
int(lnx)^2/x^3dx=-1/(4x^2)(2(lnx)^2+2lnx+1)
int(lnx)^2/x^3dx=-((2(lnx)^2+2lnx+1))/(4x^2)
