Fraction of integrals with different variables

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Discussion Overview

The discussion revolves around evaluating integrals involving different variables without using trigonometric substitution. Participants explore the possibility of combining integrals and address specific integral forms, including those with constants and subscripts.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks how to evaluate a specific integral expression without using trigonometric substitution and whether it is possible to combine the integrals into one.
  • Another participant suggests using the standard integral of \(\int \frac{1}{1 + x^2} \, dx = \tanh(x)\) and proposes a non-trig substitution to evaluate the integrals separately.
  • Some participants challenge the correctness of the integral evaluation, debating whether the integral is \(\tan(x) + C\) or \(\tanh(x) + C\), with further clarification that it is actually \(\atan(x) + C\).
  • There is a light-hearted acknowledgment of confusion regarding the integral's form, with one participant thanking another for the clarification.

Areas of Agreement / Disagreement

Participants express disagreement regarding the evaluation of the integral, particularly about the correct form of the integral involving tangent functions. The discussion remains unresolved as different interpretations are presented.

Contextual Notes

There are unresolved assumptions regarding the methods of integration and the definitions of the functions involved, which may affect the evaluation of the integrals.

huey910
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how would one evaluate this without using trig substitution? Is it possible to make one integral out of this?

{int[(y^2 + a1^2)^-1]dy +c1}/{int[(x^2 + a2^2)^-1]dx +c2} +c3

the numbers behind the 'a's and 'c's are supposed to be subscripts.

Also, how would one deal with this:

{int[(y^2 + a1^2)^-1]dy +c1}/(a tan{int[(x^2 + a2^2)^-1]dx +c2}) +c3


Please advise
 
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You can use the standard integral
[tex]\int \frac{1}{1 + x^2} \, dx = \tanh(x)[/tex]
and a "normal" (non-trig) substitution, and then you can do them separately.
 
CompuChip said:
You can use the standard integral
[tex]\int \frac{1}{1 + x^2} \, dx = \tanh(x)[/tex]
and a "normal" (non-trig) substitution, and then you can do them separately.

That doesn't seem right. Isn't that integral tan(x)+C, and not tanh(x)+C?
 
Char. Limit said:
That doesn't seem right. Isn't that integral tan(x)+C, and not tanh(x)+C?

Or rather atan(x)+C...
 
micromass said:
Or rather atan(x)+C...

Touche.
 
Heh, double fail :-)
I knew it was something with tan and an extra letter! Thanks micromass :)
 

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