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Calculus/Limits of Improper Integral

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Expert: - 8/16/2009


Question

Taking Limits
I was asked only to integrate the integral.
However I want to take one step further & determine if
convergent or divergent.

Would you please check how I took the limits.
I have never taken limits of a improper integral like
this one before.

I have solved the improper integral
sqr(1+lnx)/x lnx

My work indicates that the integral is divergent.  

Answer
OK, your integral integrates to:

2sqrt(1 + ln x) + ln([sqrt(1 + ln x) - 1]/[sqrt(1 + ln x) + 1])

(yes, you got that.)

Now at the right boundary:

x --> infinity.

First term:  2sqrt(1 + ln x) --> infinity.

What about:

ln [sqrt(1 + ln x) - 1]/[sqrt(1 + ln x) + 1]) = (I know, there is a ln)

[sqrt(1 + ln x) - 1]
--------------------- =
[sqrt(1 + ln x) + 1]


(sqrt(1 + ln x) - 1)/sqrt(ln x)
-------------------------------- =
(sqrt(1 + ln x) + 1)/sqrt(ln x)


sqrt(1/ln x + 1) - 1/sqrt(ln x)
-------------------------------
sqrt(1/ln x + 1) + 1/sqrt(ln x)

As  x -> inf,  ln x --> inf,  1/ln x --> 0, 1/sqrt(ln x) --> 0

sqrt(1/ln x + 1) - 1/sqrt(ln x)     sqrt(1) - 0
------------------------------- --> ------------ = 1
sqrt(1/ln x + 1) + 1/sqrt(ln x)     sqrt(1) + 0  

and ln(that) --> 0

So you have the right boundary --> infinity.

That does it, I am afraid.  If EITHER boundary diverges, you do not have qa convergent integral.

At the left boundary, (x->1) you have:

First term:   2sqrt(1 + ln x) --> 2,

and the second term:

ln [sqrt(1 + ln x) - 1]/[sqrt(1 + ln x) + 1]) -->

ln [sqrt(1 + 0) - 1]/[sqrt(1 + 0) + 1])

This approaches  -infinity.

So your work looks OK, but I think you could quit as soon as one boundary diverges.  

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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