Does the Improper Riemann Integral Converge or Diverge for p<1?

In summary, the improper integral \int_2^\infty \left(\frac{1}{x\log^2x}\right)^p \, dx converges for p=1 and diverges for p > 1. To show this, it is dominated by the p=1 integrand. However, it will also diverge for p < 1, as the integrand is greater than 1/x. While for p=0.9, it is mostly less than 1/x, there will be some X where the integrand is always greater than 1/x.
  • #1
tjkubo
42
0
I know that the improper integral
[itex]
\int_2^\infty \left(\frac{1}{x\log^2x}\right)^p \, dx
[/itex]
converges for p=1, but does it diverge for p>1? How do you show this?
 
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  • #2
It will converge for p > 1, since it is dominated by p=1 integrand. It will diverge for p < 1, since integrand > 1/x.
 
Last edited:
  • #3
Whoops, I meant to ask whether it diverges for p<1.

Mathman, the integrand is not always > 1/x on (2,∞) for all p<1. For p=0.9, for example, it's < 1/x for the most part.
 
  • #4
tjkubo said:
Whoops, I meant to ask whether it diverges for p<1.

Mathman, the integrand is not always > 1/x on (2,∞) for all p<1. For p=0.9, for example, it's < 1/x for the most part.
To be precise, there will be some X so that for all x > X, the integrand is > 1/x. (That is for the most part).
 

What is an Improper Riemann Integral?

An Improper Riemann Integral is a type of integral used in calculus to evaluate the area under a curve that does not have a finite limit at one or both of its bounds. It is used when the integrand is either unbounded or undefined at one or both of the bounds of integration.

How is an Improper Riemann Integral different from a regular Riemann Integral?

An Improper Riemann Integral differs from a regular Riemann Integral in that it allows for the evaluation of integrals that would otherwise be impossible using the traditional Riemann Integral. It takes into account the behavior of the integrand at the bounds of integration, rather than just the values of the integrand at each point within the interval.

What are some common examples of functions that require the use of an Improper Riemann Integral?

Functions that are discontinuous or unbounded at one or both of the bounds of integration, such as the natural logarithm or the inverse tangent, often require the use of an Improper Riemann Integral for evaluation. Other examples include functions with infinite limits, such as the Gaussian function or the exponential function.

How is an Improper Riemann Integral evaluated?

The evaluation of an Improper Riemann Integral involves taking the limit of a regular Riemann Integral as one or both of the bounds of integration approach infinity or negative infinity. This limit is then evaluated using standard techniques of integration, such as substitution or integration by parts.

What are some real-world applications of the Improper Riemann Integral?

The Improper Riemann Integral has applications in various fields of science and engineering, such as physics, economics, and electrical engineering. It is used to solve problems involving infinite series, probability distributions, and differential equations, among others.

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