How Do You Find the Anti-Derivative of (20/(1+x^2))^2?

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Discussion Overview

The discussion revolves around finding the anti-derivative of the function \((20/(1+x^2))^2\). Participants explore different approaches and interpretations of the anti-derivative, including the use of substitution and the application of derivative rules.

Discussion Character

  • Mathematical reasoning, Debate/contested, Homework-related

Main Points Raised

  • One participant expresses confusion about the anti-derivative of \((20/(1+x^2))^2\) and questions the correctness of the proposed result \(200\arctan(x)+(200x)/(x^2+1)\).
  • Another participant asserts that the result is incorrect, stating that the coefficient should be 10 instead of 200, based on the derivatives of \(\arctan(x)\) and \(x/(x^2+1)\).
  • A different approach is suggested involving the substitution \(x=\tan(y)\), leading to a transformation of the integral that simplifies the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correctness of the anti-derivative. There are competing views regarding the coefficients and methods used to arrive at the solution.

Contextual Notes

Some assumptions regarding the integration techniques and the application of derivative rules are not fully explored, leaving room for interpretation and potential errors in the proposed solutions.

calculushelp
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I just don't get it.

how does

the anti derivative of

( (20)/(1+x^(2)) )^(2)

=

200arctan(x)+(200x)/(x^(2)+1)
 
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oh p.s. that is how make TI-89 works it out.
 
calculushelp said:
I just don't get it.

how does

the anti derivative of

( (20)/(1+x^(2)) )^(2)

=

200arctan(x)+(200x)/(x^(2)+1)

It doesn't. That 200 is wrong.

The derivative of arctan(x) is 1/(x^2+ 1). The derivative of x/(x^2+ 1), using the quotient rule, is [(1)(x^2+1)- (x)(2x)]/(x^2+ 1)^2= (1- x^2)/(x^2+1)^2

Their sum is (x^2+ 1)/(x^2+ 1)^2+ (1- x^2)/(x^2+1)^2= 2/(x^2+1)^2. Multiplying by 10, not 200, would give that "20".
 
To get the antiderivative, you could substitute x=tan y and so dx=(sec y)^2 and thus simplifying
you would end up with (400/sec^2 y) dy
=>(400cos^2 y) dy
=>400(1+cos2y)dy
and integrate this and in the end substitute y=arctan x to get your answer.
 

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