Is the Integral of dx/(1-x^2) Equal to x/(1-x)?

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Discussion Overview

The discussion centers around the evaluation of the integral of dx/(1-x^2) and whether it is equal to x/(1-x). Participants explore different methods of integration and verification of the result.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests using the substitution x = sin(u) to evaluate the integral.
  • Another participant argues that the original claim is incorrect and proposes using partial fractions to decompose 1/(1 - x^2), leading to a different integral result involving natural logarithms.
  • A further reply emphasizes the importance of differentiating the proposed solution to verify its correctness, concluding that the differentiation does not yield the expected result, thus indicating the original claim is false.

Areas of Agreement / Disagreement

Participants do not agree on the correctness of the original integral claim, with multiple competing views presented regarding the evaluation of the integral.

Contextual Notes

Some participants rely on different methods of integration and verification, which may depend on specific assumptions or definitions related to the integral and its evaluation.

gigi9
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Calculus help please! another one

Can someone please show me how to show that this problem below is CORRECT? Thanks a lot.
Integral of dx/(1-x^2) = x/(1-x)
 
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Use the substitution x=sinu.
 


Hello, gigi9!

The answer is not correct . . .

Originally posted by gigi9
Can someone please show me how to show that this problem below is CORRECT?
Integral of dx/(1-x^2) = x/(1-x)
We can use Partial Fractions: 1/(1 - x^2) = A/(1 - x) + B/(1 + x)
and find that: A = 1/2, B = -1/2.

The answer will be: (1/2) ln|(1 - x)/(1 + x)| + C
 
Can someone please show me how to show that this problem below is CORRECT? Thanks a lot.
Integral of dx/(1-x^2) = x/(1-x)

The easiest way to check if an integral is correct is to invert the operation and differentiate.

Via the fundamental theorem of calculus, if

∫ dx / (1 - x^2) = x / (1 - x)

then

1 / (1 - x^2) = (d/dx) (x / (1 - x))

So if we actually perform the differentiation, we get:

1 / (1 - x^2) = 1 / (1 - x)^2

Because this equation is false, the original problem (as you've written it) must be false as well.
 

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