Integration with Trigonometric Substitution

[dollarbills10]

Hi,

I am seeking some input for an integral I have been stumped on for a few days.

This is the integral:

[(a^2 - s^2)^1/2]/(x-s) ds evaluated over the bounds from -a to a. The symmetry of the integration area allows the integral to be evaluated from 0 to a, and doubled.

I have always been conditioned to use trig substitution for an integral when an expression such as (or usually, exactly as) (a^2 - s^2)^1/2 is in the integrand. However, this is unique in that this expression is being divided by x-s in the integrand.

Any suggestions?

Thank you in advance, I am new so I apologize if I have not presented my topic in the most ideal format.

 

[mathman]

The term (x-s) destroys the symmetry.

 

[dollarbills10]

Hi,

Mathman, thank you for your comment.

I have attached a much better pictorial of the integrand in discussion. I have failed in trying to decipher the latex language in the forum.

I have also included my trig substitution. However, as I have hinted, I have no clue as to where to go from here.

Any comments appreciated.

 

[mathman]

I have never seen this before. Be careful if |x| < a.

 

[dollarbills10]

Mathman,

Thank you again for the comment. Obviously, the singularity is present.

After some more investigation, the solution will definitely follow that of Cauchy Principal Value integrals, in as much as the Residue Theorem and the poles of the integral in the upper half of a complex plane are used to evaluate the integral. However, I believe you can still use trig substitutions, but it may not be necessary. If interested, I'll keep updating.

Thanks.

 

[paulfr]

However, as I have hinted, I have no clue as to where to go from here.
Any comments appreciated.

Whenever you have a rational trigonometric integrand, The Wieirstrass Substitution Method [also called the Tangent Half Angle Method] is a powerful technique to use. It converts the rational trig functions to rational algebraic functions so that one can then use Partial Fractions or U Substitution or Long Division or Complete the Square methods.

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

Here is a sample problem showing how it works.
Click on "Show Steps" in the upper right corner to see the details of the solution.

http://www.wolframalpha.com/input/?i=integral+[+sin+x+%2B+cos+x+]+%2F+[+1+%2B+cos+x+]+dx

You may need to cut and paste in the link

It is not as difficult as it looks once you understand what is going on between "phi" and "phi / 2"

Enjoy