Let
y=(lnx)^2 /(x^3)y=(lnx)2x3
intydx=int(lnx)^2 /(x^3)dx∫ydx=∫(lnx)2x3dx
Let
t=lnxt=lnx
e^t=xet=x
e^2t=x^2e2t=x2
t^2=(lnx)^2t2=(lnx)2
(dt)/dx=1/xdtdx=1x
dt=1/xdxdt=1xdx
intydx=int(lnx)^2 /(x^2)(1/xdx)∫ydx=∫(lnx)2x2(1xdx)
=intt^2/e^(2t)dt=∫t2e2tdt
intydx=intt^2e^(-2t)dt∫ydx=∫t2e−2tdt
integrating by parts
intudv=uv-intvdu∫udv=uv−∫vdu
u=t^2u=t2
du=2tdtdu=2tdt
dv=e^-2tdtdv=e−2tdt
v=-1/2e^(-2t)v=−12e−2t
intt^2e^(-2t)dt=t^2(-1/2e^(-2t))-int(-1/2e^(-2t))(2tdt)∫t2e−2tdt=t2(−12e−2t)−∫(−12e−2t)(2tdt)
=-t^2/2e^(-2t)+intte^(-2t)dt=−t22e−2t+∫te−2tdt
=-t^2/2e^(-2t)+I_1=−t22e−2t+I1
where
I_1=intte^(-2t)dtI1=∫te−2tdt
integrating by parts
intudv=uv-intvdu∫udv=uv−∫vdu
u=tu=t
du=dtdu=dt
dv=e^(-2t)dtdv=e−2tdt
v=-1/2e^(-2t)v=−12e−2t
intte^(-2t)dt=t(-1/2e^(-2t))-int(-1/2e^(-2t))dt∫te−2tdt=t(−12e−2t)−∫(−12e−2t)dt
=-1/2te^(-2t)+1/2(-1/2)e^(-2t)=−12te−2t+12(−12)e−2t
intte^(-2t)dt=-1/2te^(-2t)-1/4e^(-2t)∫te−2tdt=−12te−2t−14e−2t
I_1=-1/2te^(-2t)-1/4e^(-2t)I1=−12te−2t−14e−2t
intt^2e^(-2t)dt=-t^2/2e^(-2t)+I_1∫t2e−2tdt=−t22e−2t+I1
intt^2e^(-2t)dt=-t^2/2e^(-2t)+(-1/2te^(-2t)-1/4e^(-2t))∫t2e−2tdt=−t22e−2t+(−12te−2t−14e−2t)
intt^2e^(-2t)dt=-t^2/2e^(-2t)-1/2te^(-2t)-1/4e^(-2t)∫t2e−2tdt=−t22e−2t−12te−2t−14e−2t
intt^2e^(-2t)dt=-1/4(2t^2+2t+1)e^-2t∫t2e−2tdt=−14(2t2+2t+1)e−2t
Replacing
t=lnxt=lnx
e^(-2t)=1/x^2e−2t=1x2
int(lnx)^2/x^2(1/x)dx=-1/4(2(lnx)^2+2lnx+1)(1/x^2)∫(lnx)2x2(1x)dx=−14(2(lnx)2+2lnx+1)(1x2)
int(lnx)^2/x^3dx=-1/(4x^2)(2(lnx)^2+2lnx+1)∫(lnx)2x3dx=−14x2(2(lnx)2+2lnx+1)
int(lnx)^2/x^3dx=-((2(lnx)^2+2lnx+1))/(4x^2)∫(lnx)2x3dx=−(2(lnx)2+2lnx+1)4x2
