Need help with comparison method for improper integral?

In summary, when using the comparison method for the integral of arctan(x) over 2+e^x, it is suggested to compare it to the function 2/(2+e^x) as tan-1(x) < 2 for all x. This avoids any issues with division by zero and can be used to determine the convergence or divergence of the integral. Additionally, if the comparison test is used, it is important to show that the chosen function is always greater than or equal to the original integrand.
  • #1
mottov2
13
0

Homework Statement


[itex]\int \frac{arctan(x)}{2+e^x}dx[/itex]
where the interval of the integrand is from 0 to infinity.

In order to use the comparison method I need to compare 2 functions but I am having so much difficulty figuring out what function to compare it to.

Its not just this particular question, any question asking to use comparison method, I can't seem to figure out the function to compare it to.

Can I get some tips to finding a function to compare it to?
 
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  • #2
tan-1(x) < 2 for all x, so therefore...

[tex]\frac{tan^{-1}(x)}{2 + e^{x}} < \frac{2}{2 + e^{x}}[/tex]

for all x (Since 2+ex is never 0, one doesn't have to worry about division by zero.)

So try using the comparison test with that.
 

Related to Need help with comparison method for improper integral?

1. What is an improper integral?

An improper integral is an integral where at least one of the limits of integration is infinite, or the integrand function is not defined at some point in the interval of integration.

2. Why do we need a comparison method for improper integrals?

The comparison method allows us to determine the convergence or divergence of an improper integral by comparing it to a known integral that has a known convergence or divergence behavior. This makes it easier to evaluate the integral and determine its convergence or divergence.

3. How does the comparison method work?

The comparison method involves comparing the given integral to another integral that is either smaller or larger, and whose convergence or divergence behavior is known. If the given integral is smaller than the known integral and the known integral converges, then the given integral also converges. If the given integral is larger than the known integral and the known integral diverges, then the given integral also diverges.

4. What are some common known integrals used in the comparison method?

Some common known integrals used in the comparison method are the p-series, geometric series, and harmonic series. These series have well-known convergence or divergence behavior and can be used to compare with other improper integrals.

5. Are there any limitations to the comparison method?

Yes, there are limitations to the comparison method. It can only be used for improper integrals where the integrand function is positive and continuous on the interval of integration. It also cannot be used for integrals with infinitely oscillating functions or integrals with non-integrable singularities.

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