Evaluate difficult integral

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In summary, this student is trying to solve an integral equation and does not know how to begin. He or she is looking for help from others and asks for suggestions on how to start.
  • #1
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Homework Statement



I want to evaluate the following integral:

[itex]\int \frac{\sqrt{a^2-x^2}(a-x)^{-1/c}(x-d)}{x^2-b^2}dx[/itex]

Homework Equations





The Attempt at a Solution



I rewrote it in the following form (to try to make it simpler):

[itex]\int \frac{\sqrt{a+x}(a-x)^{1/2-1/c}(x-d)}{x^2-b^2}dx[/itex]

I have no idea how to start integrating this. I have tried putting it into Mathematica, but Mathematica didn't evaluate it.
Any ideas will be very much appreciated.
 
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  • #2
We know some things about the integrand;
It has singularities at [itex]\pm b[/itex].
If c is positive, there is a singularity at x=a.
It is complex for x>a.

:yuck: If this were a definite integral, then I would suggest some kind of contour integration approach. But actually finding an antiderivative for that? Good luck, friend.
 
  • #3
If this integral was a definite integral, with the integration limits being:
d (for the lower limit), and f (for the upper limit);

would this be able to integrate using contour integration? If so, do you have any hints how I can go through that approach?

Thank you.
 
  • #4
Possibly. Contour integration methods for computing real integrals are really a bit of an art. There is no general procedure; the way you do it depends on what your real integral is. Also, in your case, it also depends on what your lower limit d, and upper limit f, are with respect to the singularities mentioned.

If you haven't learned any complex analysis (eg. Cauchy-Goursat theorem, Residue Theorem, Cauchy integral formula, etc.), then there really is no way I can guide you to attempt to solve this problem in only a few lines. So I will first recommend you read the following wikipedia article, to get a general sense of how the procedure works.

http://en.wikipedia.org/wiki/Methods_of_contour_integration#Applications_of_integral_theorems

Then, if you feel up to it, take out any decent textbook on the subject of Complex Analysis from your library, and it should have a section on the evaluation of Real integrals by contour integration.

I should warn you that I personally have never tried working out an integral as messy as the one you have. In theory, you should be able to do it, I just don't have the time or drive to try. Complex analysis is very useful, and the theoretical results are beautiful, but the applications (like this) are tedious, and at times very difficult.

Good luck.
 

1. How do you approach solving a difficult integral?

When faced with a difficult integral, it is important to first identify the type of integral it is (such as trigonometric, exponential, or logarithmic). Then, you can use various techniques such as substitution, integration by parts, or trigonometric identities to simplify the integral into a more manageable form.

2. What is the importance of evaluating difficult integrals?

Difficult integrals often arise in real-world applications, particularly in physics and engineering. By being able to evaluate these integrals, scientists are able to solve complex problems and make accurate predictions about the behavior of systems.

3. Can a difficult integral be solved using numerical methods?

Yes, difficult integrals can also be solved using numerical methods such as the trapezoidal rule or Simpson's rule. These methods involve approximating the integral using a series of smaller, simpler integrals.

4. Are there any general strategies for evaluating difficult integrals?

Yes, there are a few general strategies that can be helpful when evaluating difficult integrals. These include recognizing patterns, using symmetry when applicable, and breaking the integral into smaller parts.

5. How do you know if an integral is too difficult to solve analytically?

It can be difficult to determine the complexity of an integral without attempting to solve it. However, some indicators that an integral may be too difficult to solve analytically include the presence of multiple variables, complicated functions, or a lack of known integration techniques for the type of integral.

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