How to test if a definite integral is finite or not?

In summary, the conversation discusses the difficulty of evaluating a complicated integral and how to determine if the result is finite or not. It is suggested that there may be certain conditions on the function involved, such as the integrand approaching zero as x approaches infinity. However, this condition is not sufficient and it is also mentioned that a function with a greater exponent than 1/x may have a finite integral. The example integral given, \int_{-\infty}^{\infty} e^{-2a \tanh^2(bx)} dx, is shown to be not finite using this reasoning.
  • #1
arroy_0205
129
0
Suppose I have a complicated integral whose exact evaluation seems extremely difficult or may be even impossible, in such a case is there any way to tell if the integration result is finite or not? suppose the problem is
[tex]
\int_{-\infty}^{\infty} f(x;a,b) dx
[/tex]
I think there might be some conditions on the function involved. Actually the function contains some parameters also (a,b) which can be taken to be constants for a particular case. Now I am looking for a condition general enough to handle arbitrary parameters, ie, can I tell if the integral is finite for any arbitrary values of those parameters? If yes, then how or under what condition? Take as an example:
[tex]
\int_{-\infty}^{\infty} e^{-2a \tanh^2(bx)} dx
[/tex]
Is this finite?
 
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  • #2
I may be a little rusty. If I remember correctly, |tanh(bx)| -> 1 as x -> oo. In which case, the integral certainly is NOT finite. A necessary condition for integral to be finite with an infinite domain is the integrand go to zero as x -> oo. Also, this condition is not sufficient!
 
  • #3
If i remember correctly, besides using definition you can use the fact that any positive (or negative) function has a finite integral, if its falling with a greater exponent than 1/x. (equivalently - if its below x^-(1+eps) for any x greater than some x0). It doesn't work ok for an alternating function.
This applies if you are interested in a true integral, not prime value. Also, this covers just what happens when x goes to infinity, not points where function itself diverges.
 

1. How do you know if a definite integral is finite?

The definite integral is finite if the function being integrated is continuous on the interval of integration and if the interval is bounded. This means that the function does not have any breaks or jumps in its graph and that the interval of integration has a defined starting and ending point.

2. What is the difference between a finite and infinite definite integral?

A finite definite integral has a defined numerical value, while an infinite definite integral does not. This means that the area under the curve can be calculated for a finite integral, but not for an infinite integral.

3. How can you test if a definite integral is finite or infinite?

To test if a definite integral is finite, you can use various techniques such as the comparison test, the limit comparison test, or the integral test. These methods involve determining the convergence or divergence of the integral by comparing it to a known function or using the properties of integrals.

4. Can a definite integral be both finite and infinite?

No, a definite integral can only be either finite or infinite. If the function being integrated is continuous and the interval of integration is bounded, the integral will be finite. If the function is not continuous or the interval is unbounded, the integral will be infinite.

5. What is the significance of determining if a definite integral is finite or infinite?

Determining if a definite integral is finite or infinite is important in many areas of mathematics and science. It can help us understand the behavior of functions and their relationships, as well as solve problems in calculus, physics, and engineering. Additionally, knowing if an integral is finite or infinite can help us make predictions and draw conclusions in various fields of study.

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