Expert: Paul Klarreich - 10/8/2007

Question
Umm sry Mr Klarreich, but i typed that last problem wrong. See it wasnt as easy as the one I gave you (otherwise I would have got it). THe real one is this:

{     dx
|----------
}x(log x)^(5/9)

again with the log with base 8.

Thanks again

o yeah Im only a junior in high school so...im not in college yet

Answer
Questioner:   Jeff
Category:  Calculus
Private:  No
 
Subject:  integration
Question:  Umm sry Mr Klarreich, but i typed that last problem wrong. See it wasnt as easy as the one I gave you (otherwise I would have got it). THe real one is this:

{     dx
|----------
}x(log x)^(5/9)

again with the log with base 8.

Thanks again

Oh yeah, I'm only a junior in high school so...im not in college yet
 
===============================
Hi again, Jeff,

This example:

{     dx
| -----------------
}  x (log x)^(5/9)

isn't really much different.  Keeping in mind that:

log8(x) = ln(x)/ln(8)

you have:

{         dx
| ----------------------
}  x (ln(x)/ln(8))^(5/9)

{  [ln(8)]^(5/9)  dx
| ----------------------
}  x [ln(x)]^5/9

and the [ln(8)]^(5/9)  is a constant factor.  I'll call it L8.

   {         dx
L8  | ----------------------
   }  x [ln(x)]^5/9

Now do the same stuff:  u = ln x,  du = dx/x


  {    du
L8 | --------
  }  u^5/9

  {
L8 | u^-5/9 du
  }

  u^4/9
L8 -------
   4/9

9   
---- [ln(8)]^(5/9) [ln x]^4/9
4