Finding Whether Improper Integrals Converge

  • Thread starter rocapp
  • Start date
  • Tags
    Integrals
In summary, an improper integral is an integral with infinite limits or a discontinuous integrand. To determine if it converges, tests such as the limit comparison, comparison, and integral tests can be used. A convergent improper integral has a finite value, while a divergent one does not. However, it is possible for an improper integral to have a finite value and still be divergent due to oscillations. These integrals have various real-life applications in fields like physics, engineering, and economics. They can be used to calculate energy, fluid flow, and profits over time.
  • #1
rocapp
95
0
Hi all,

I'm having trouble with finding an improper integral.

The problem is ∫10(xln(x))dx

My book says the answer is -1/4, but I do not understand how this is the case.

lim(xlnx) as x approaches 1- = 0

lim(xlnx) as x approaches 0+ = ∞

So how does this value converge at -1/4?

Thanks in advance!
 
Physics news on Phys.org
  • #2
rocapp said:
lim(xlnx) as x approaches 0+ = ∞

What makes you say that? Did you try Lhopital?
 
  • #3
Ah. I see that. So now I just find the integral and from zero to one, correct?
 

FAQ: Finding Whether Improper Integrals Converge

What is an improper integral?

An improper integral is a type of integral where one or both of the limits of integration are infinite or the integrand is not defined for some points in the interval.

How do you determine if an improper integral converges?

To determine if an improper integral converges, you can use the limit comparison test, comparison test, or the integral test. These tests involve taking the limit of the integral as the limits of integration approach infinity or a point of discontinuity.

What is the difference between a convergent and a divergent improper integral?

A convergent improper integral is one that has a finite value when the limit of integration is taken to infinity or a point of discontinuity. On the other hand, a divergent improper integral is one that does not have a finite value in these cases.

Can an improper integral have a finite value but still be divergent?

Yes, it is possible for an improper integral to have a finite value but still be divergent. This can occur when the integral oscillates between positive and negative values, resulting in a cancellation of the infinite parts.

What are some real-life applications of improper integrals?

Improper integrals have many real-life applications in fields such as physics, engineering, and economics. For example, they can be used to calculate the total energy of an object, the amount of fluid flowing through a pipe, or the total profit of a business over time.

Similar threads

Improper integral convergence from 0 to 1
Replies
13
Views
2K
Finding if an improper integral is Convergent
Replies
5
Views
1K
What is the Correct Integral for Finding the Area Below y=0 and Above y=lnx?
Replies
5
Views
1K
Prove limit comparison test for Integrals
Replies
2
Views
1K
Help with proving that Improper Integral is Divergent
Replies
9
Views
2K
Generalize improper integral help
Replies
1
Views
1K
Prove that this series converges uniformly on R
Replies
4
Views
996
How to show that these sequences are summable?
Replies
1
Views
1K
Improper Integral: ∫(sin(x)+2)/x^2 from 2 to ∞ - Converge or Diverge?
Replies
4
Views
1K
Does the series converge using the integral test?
Replies
16
Views
3K

Hot Threads

Recent Insights

Back
Top