20090719, 22:36  #1 
Dec 2008
7^{2}·17 Posts 
Integral Variation
I was looking at an integral the other day:
. And I was told that it varies as 1.5x/log x, because I was trying to show that: , where can be taken as 27. I mean, I've taken BC Calc two years ago, I took Linear Algebra last year, and I will take Differential Equations in the coming fall, but I have never heard of the term "vary" when dealing with integrals, or perhaps I know of a term with a similar definition. Would anyone mind explaining to me what it means that the integral varies as 1.5x/log x? (and/or give me a useful link talking about it). 
20090720, 06:52  #2  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
The variable is inside the integral. The variable is outside the integral.
Let . The claim is that varies as 1.5x/log x. Quote:
happens to have an integral inside it, but that's not important for "varies". The function varies. The function has an integral inside it. Those are two separate statements. Combining them into "the integral varies" is just a contraction that happens to be okay because the function has only the integral in it (no other terms), so referring to the function as "the integral" is permissible here. If the function were the sum of two different integrals (presumably both with an upper limit related to x), then one would need to say, "the sum of integrals varies as ..." Last fiddled with by cheesehead on 20090720 at 07:02 

20090720, 12:58  #3  
Nov 2003
2^{3}·7^{2}·19 Posts 
Quote:
If "varies" means "is asymptotic to", then the claim is wrong. int 1/log^3 t dt = t/(2 log t)  t/(2 log^2 t) + li(t)/2. := g(t) Now evaluate g(x^1/3)  g(2). 

20090720, 14:24  #4 
Dec 2008
7^{2}·17 Posts 

20090720, 15:57  #5 
"Kyle"
Feb 2005
Somewhere near M50..sshh!
2·3·149 Posts 
Apparently I am rusty on my calculus as well. I have been through all the basic calculus classes (13) but I cannot seem to remember how to integrate that before you plug in your bounds. It is not possible to make a usubstitution either of u=log x or 1/log x... The only thing I can think of is integration by parts  tabular method but I do not see how this would work when you have one term that will never differentiate to zero. Dr. Silverman, how did you get this answer?

20090720, 18:16  #6  
Nov 2003
2^{3}×7^{2}×19 Posts 
Quote:
Integration by parts combined with the definition of li(x). [li(x) is not an elementary function] 

20090720, 20:00  #7  
"Kyle"
Feb 2005
Somewhere near M50..sshh!
2·3·149 Posts 
Quote:
I apologize, this is not my homework problem but I am always eager to learn something new 

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