Integrating (ln x)^2/e^5t in Variation of Parameters Problem

[eagleswings]

i have to integrate (ln x)(ln x)/ e^5t [in a variation of parameters problem] and have looked everywhere to see if Ln x multiplied by itself can be shortened to something like ln x^2 or some other reasonable thing but can't find such a rule anywhere. do i have to do this the long way with integration by parts? not even sure what to do with three things in integration by parts.

 

[nicksauce]

Is it e^5t or e^5x?. If it's the latter I don't believe there's a closed form solution, but if it's the former, then it's more doable.

Let u = lnx, du= dx/x, dv=lnxdx, v = xlnx - x (Verify this by IBP). Then all the integrals are doable.

And yes, there are no identities to simplify ln(x)^2.

 

[eagleswings]

yes actually it is e^5x. all the t's that go into the integral have to change to x. but perhaps i can start with what you gave me - thanks!

 

[nicksauce]

You can see what the integral is here
http://integrals.wolfram.com/index.jsp?expr=ln(x)*ln(x)*exp(-5x)&random=false

But it's not very pretty.

 

[eagleswings]

man, i can'te even read that one. it's got symbols i haven't seen used, maybe they are just variables, but unusual ones. maybe i learn that next year! sigh.