How to solve an Improper Integral of Type 2?

In summary, the conversation discusses the steps to calculate an improper integral of type 2, which involves setting the limits, putting it in integral form, and using the FTC. An example is given to better understand the process, which involves using the theorem of addition and computing two limits. The conversation also clarifies that the limits would be 1+ and 1- in this particular example.
  • #1
cmab
32
0
I've posted on Homeworks one of the number I did not understand. However, I would like to know the steps to calculate an improper integral of type 2. The type 2 is the one from constant a to constat b, not the one with inifnite.

Please tell me the steps the accomplish it. :smile:

I know that I'm supposed to set the limits, then put it in the integral form, than FTC it. But, I'm not sure.
 
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  • #2
Could you give example so I could understand question a little better.
 
  • #3
limitapproaches0 said:
Could you give example so I could understand question a little better.


Let's say something like [int a=0 b=2] 1/(x-1)^(1/3) dx
 
  • #4
Use the theorem of addition

[tex] \int_{0}^{2} f(x) \ dx=\int_{0}^{1} f(x) \ dx+\int_{1}^{2} f(x) \ dx [/tex]

You have to integrate & compute 2 limits...

Daniel.
 
  • #5
dextercioby said:
Use the theorem of addition

[tex] \int_{0}^{2} f(x) \ dx=\int_{0}^{1} f(x) \ dx+\int_{1}^{2} f(x) \ dx [/tex]

You have to integrate & compute 2 limits...

Daniel.

The limits would be 1+ and 1- right ?
 
  • #6
It should be

[tex] \lim_{a\nearrow 1}\int_{1}^{a} f(x) \ dx [/tex]

[tex] \lim_{b\searrow 1}\int_{b}^{2} f(x) \ dx [/tex]


Daniel.
 

Related to How to solve an Improper Integral of Type 2?

1. What is an Improper Integral of Type 2?

An Improper Integral of Type 2 is an integral where one or both of the limits of integration are infinite, or when the integrand function is unbounded at one or more points within the limits of integration.

2. How do I determine if an integral is of Type 2?

If the integral has one or both limits of integration as infinity, or if the integrand function has a vertical asymptote within the limits of integration, then it is an Improper Integral of Type 2.

3. What is the process for solving an Improper Integral of Type 2?

The process for solving an Improper Integral of Type 2 involves first rewriting the integral in terms of a new variable, then determining if the integral converges or diverges using a convergence test, and finally evaluating the integral using appropriate techniques such as integration by parts or substitution.

4. What is the significance of solving an Improper Integral of Type 2?

Solving an Improper Integral of Type 2 allows for the calculation of the area under a curve even when the limits of integration are infinite or the integrand function is unbounded at certain points. This is important in many applications in physics, engineering, and other fields.

5. Are there any common mistakes to avoid when solving an Improper Integral of Type 2?

One common mistake is to forget to use appropriate techniques, such as integration by parts, when evaluating the integral. It is also important to carefully check the convergence of the integral using convergence tests, as well as the limits of integration to ensure that the integral is indeed of Type 2.

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