Integral evaluated at +/- infinity

In summary, the conversation discusses a physics problem involving an integral and the final step of evaluating it at infinity. The person is unsure of how to evaluate this step and is seeking clarification. They realize that their approach may have been sloppy and that improper integrals involve limits.
  • #1
diffusion
73
0
Ok, I'm trying to solve this physics problem and I've come to the following integral ([tex]d[/tex] is taken to be some constant):

1. [tex]\int^{+\infty}_{-\infty}{\frac{1}{(x^2 + d^2)^\frac{3}{2}}}dx[/tex]

Now, integrating this I am supposed to get

2. [tex]{\frac{x}{d^2\sqrt{x^2 + d^2}}}[/tex], evaluated at [tex]\pm\infty[/tex] (Sorry, don't know latex code for that).

I don't know how to evaluate this last step. It would seem I should either get 0 or infinity, but apparently that isn't the answer. Could anyone enlighten me?
 
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  • #2
This is one reason why sloppiness can be bad. If you weren't thinking of it as "evaluate at infinity", you would have remembered that such improper integrals are limits... and you would have known to apply what you know about limits!
 
  • #3
For large magnitude x, your expression (2.) is approx. x/[d2|x|]. You should be able to finish.
 

1. What is an integral evaluated at +/- infinity?

An integral evaluated at +/- infinity is a mathematical concept where the limits of integration extend to positive or negative infinity. This means that the function is being evaluated over an infinite range of values.

2. How is an integral evaluated at +/- infinity different from a regular integral?

An integral evaluated at +/- infinity is different from a regular integral because it involves evaluating the function over an infinite range, while a regular integral has finite limits of integration.

3. Can an integral evaluated at +/- infinity have a finite value?

Yes, an integral evaluated at +/- infinity can have a finite value if the function being integrated approaches zero as the limits of integration approach infinity.

4. What is the significance of evaluating an integral at +/- infinity?

Evaluating an integral at +/- infinity can help determine the behavior and properties of a function over a large range of values. It is also commonly used in physics and engineering to solve problems involving infinite systems.

5. How do you evaluate an integral at +/- infinity?

To evaluate an integral at +/- infinity, the function is first simplified using algebraic techniques such as substitution or partial fractions. Then, the limits of integration are replaced with positive or negative infinity, and the resulting function is evaluated using appropriate integration techniques such as integration by parts or trigonometric substitution.

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