How can I integrate Sqrt[x^2-a] using trig substitutions?

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Discussion Overview

The discussion revolves around the integration of the function Sqrt[x^2-a] using trigonometric substitutions. Participants explore the implications of the parameter 'a' being either positive or negative and how it affects the integration process.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant requests assistance with the integration, noting that 'a' could be positive or negative, which complicates the use of sqrt(a) in the answer.
  • Another participant argues that if a>0, then sqrt(a) can be included in the answer, while for a<0, the integrand transforms to sqrt{x^2+b} where b=-a.
  • Another viewpoint suggests that the sign of 'a' should not affect the integral, implying that the integral remains the same regardless of whether the result is imaginary.
  • Multiple participants recommend using trigonometric substitutions, specifically secant or tangent, depending on the sign of 'a'.
  • One participant raises a concern about the substitution x = sqrt(a)Secy leading to an integrated result that includes a tangent function, indicating potential complications in the integration process.

Areas of Agreement / Disagreement

Participants express differing opinions on the impact of the sign of 'a' on the integration process, indicating that multiple competing views remain unresolved.

Contextual Notes

There are unresolved assumptions regarding the treatment of the parameter 'a' and its implications for the integration process, particularly concerning the use of trigonometric substitutions and the resulting expressions.

coverband
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Could someone show me how to integrate this. Bear in mind 'a' could be positive or negative thus i don't think we can use sqrt(a) in our answer...
 
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Well if a>0, then you can have sqrt(a) in your answer...

But for a<0 => -a>0 meaning that your integrand will become sqrt{x^2+b} where b=-a
 
Shouldn't matter if it's positive or negative, the integral is still the same, imaginary or not.
 
use trig substitutions, sec or tan depending on whether a is positive or negative.
 
mathwonk said:
use trig substitutions, sec or tan depending on whether a is positive or negative.


only problem is say if I use the substitution Let x = sqrt(a)Secy the integrated result has a tany in it
 

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