How to do this integral with substitution?

In summary, this integral is related to the normal probability distribution and has been studied extensively analytically and numerically.
  • #1
rwooduk
762
59
This is the integral: NB everything in the expenential is in a squared bracket, couldn't get tex to do it

[tex]\frac{1}{2\pi}\int_{\infty }^{\infty} e^{\tfrac{q\Delta}{\sqrt{2}}-\tfrac{ix}{\sqrt{2}\Delta}}dq[/tex]

The only information the tutor has given use to solve this is to use substitution and this:

[tex]\int_{\infty }^{\infty} e^{\tfrac{-q^{2}}{2\Delta^{2}}} dq = \sqrt{2 \pi}\Delta[/tex]

Please could someone give me a point int the right direction? If w is the new variable what should i put it equal to?

Thanks for any help!
 
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  • #2
rwooduk said:
This is the integral: NB everything in the expenential is in a squared bracket, couldn't get tex to do it

[tex]\frac{1}{2\pi}\int_{\infty }^{\infty} e^{\tfrac{q\Delta}{\sqrt{2}}-\tfrac{ix}{\sqrt{2}\Delta}}dq[/tex]

The only information the tutor has given use to solve this is to use substitution and this:

[tex]\int_{\infty }^{\infty} e^{\tfrac{-q^{2}}{2\Delta^{2}}} dq = \sqrt{2 \pi}\Delta[/tex]

Please could someone give me a point int the right direction? If w is the new variable what should i put it equal to?

Thanks for any help!

These integrals do not have solutions composed of elementary functions, and hence are not amenable to solution by substitution, except to get them into one of the integral forms discussed in the articles below.

They are related to the normal probability distribution and thus have been studied extensively analytically and numerically.

You might want to consult these articles:

http://en.wikipedia.org/wiki/Normal_distribution

http://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions
 
  • #3
rwooduk said:
This is the integral: NB everything in the expenential is in a squared bracket, couldn't get tex to do it

[tex]\frac{1}{2\pi}\int_{\infty }^{\infty} e^{\tfrac{q\Delta}{\sqrt{2}}-\tfrac{ix}{\sqrt{2}\Delta}}dq[/tex]
Are you missing a sign on the lower limit of integration?
Also, what is ##\Delta##? Since the variable of integration is q, then presumably ##\Delta## and x are to be treated as constants.

What do you mean by "squared bracked"? Some people call these -- [ ] -- square brackets. Did you mean that the quantity in brackets is squared, like this -- [ ... ]2?

 
  • #4
The exponent in the original integral looks completely screwed up. Rewrite it.
 
  • #5
People won't stop bombarding this forum with nonelementary integrals, won't they? :D
 

1. How do I choose the right substitution for an integral?

The key to choosing the right substitution for an integral is to look for a part of the integrand that resembles a function whose derivative is also present in the integrand. This will help simplify the integral and make it easier to solve.

2. What is the purpose of substitution in integration?

The purpose of substitution in integration is to simplify the integrand by replacing a complicated expression with a simpler one. This allows us to solve integrals that would otherwise be difficult or impossible to solve.

3. How do I know if I need to use substitution in an integral?

If you encounter an integral that contains a composition of functions (such as f(g(x))), it is likely that you will need to use substitution to solve it. Additionally, if the integrand contains a function and its derivative, substitution may also be necessary.

4. What are the steps for using substitution in integration?

The steps for using substitution in integration are as follows:

  1. Identify the part of the integrand that can be replaced with a new variable.
  2. Choose a suitable substitution for the identified variable.
  3. Use the substitution to rewrite the integrand in terms of the new variable.
  4. Solve the resulting integral using techniques such as u-substitution or integration by parts.
  5. Finally, substitute the original variable back into the solution to get the final answer.

5. Are there any common mistakes to avoid when using substitution in integration?

One common mistake to avoid is forgetting to substitute the limits of integration when using substitution. It is important to replace the original variable in both the integrand and the limits before solving the integral. Additionally, be sure to check your answer by differentiating it to ensure that it matches the original integrand.

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