Question

Can this be integrated:

ln t /[t * e^(4t) ]

Answer 1

This isn't really an answer but my idea would be to write it as (ln t/t)*e(-4t). The first part is in the form f'(x)*f(x) so it can be integrated.

So integration by parts should work.

Answer 2

But even if I can integrate the first part, how does that help me? The function needs to be integrated as a whole. Maybe I'm misunderstanding what you tried to say.

Answer 3

If you write it like this, you could do integration by parts: \[\int\limits \ln(t) * (1/te ^{4t}) dt\]

Answer 4

The integration by parts formula is \[\int\limits u dv = uv-\int\limits v du\] so choose either ln(t) or (1/te^4t) to be u and the other to be dv, and use the formula. I don't know what kind of answer that gives as I haven't done it myself but that seems to be a logical approach.

Answer 5

Or if you go by what beginnersmind is saying, you could write it like that too and do integration of parts but for ln(t)/t you'd have to use the product rule inside your integration by parts.

Answer 6

Thanks, yeah that makes sense.

Answer 7

Actually the trick is that the derivative of (lnx)^2 is 2lnx/x. So the right way to separate seems to be

dv = lnx/x and u = e^(-4t)

Answer 8

Thanks to both of you, I'm gonna try it out.

Answer 9

Actually, just ignore everything I said :(
I don't think it's helpful at all...

Answer 10

I've tried it several ways now and all ways look complicated. It looks like you'd just have to keep doing integration by parts forever. There must be a trick somewhere to simplify it...

Answer 11

yeah this seems like a very complicated integral...round and round