Calculus 2: Trigonometric Substitution, using Z = tan(x/2)

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Homework Help Overview

The discussion revolves around the use of trigonometric substitution in calculus, specifically utilizing the substitution \( z = \tan\left(\frac{x}{2}\right) \). Participants are exploring the implications of this substitution on the integration process.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants express confusion about the substitution process and its application, particularly regarding the transformation of limits and the derivation of \( dx \). Some are questioning the accuracy of their substitutions and the resulting bounds for integration.

Discussion Status

The discussion is active, with participants sharing insights and clarifications about the substitution method. Some guidance has been offered regarding the need to adjust limits when changing variables, and one participant has acknowledged a misunderstanding regarding the bounds after reviewing the solution.

Contextual Notes

There appears to be a lack of clarity on the correct limits of integration when applying the substitution, as well as uncertainty about the derivation steps involved in the process. Participants are working within the constraints of homework guidelines that may limit the resources they can consult.

kiz
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Homework Statement



2e2jsc8.jpg

Homework Equations



25rlguf.jpg

The Attempt at a Solution



After substituting:

2cf7axc.jpg

Using
14wzxiw.jpg

2yo47qx.jpg

I'm stuck here:
dzlsp.jpg

I can't seem to find anything online about this substitution. Any help would be appreciated. thanks.
 
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kiz said:

Homework Statement



2e2jsc8.jpg

Homework Equations



25rlguf.jpg

The Attempt at a Solution



After substituting:

2cf7axc.jpg

Using
14wzxiw.jpg

2yo47qx.jpg

I'm stuck here:
dzlsp.jpg

I can't seem to find anything online about this substitution. Any help would be appreciated. thanks.

I am so glad I have no need to remember any of this. my computer does all the calculations for me :)
 
http://en.wikipedia.org/wiki/Tangent_half-angle_formula You seem to have turned a '2' into a 'z' in you dx derivation. And you would have to change the limits to 'z' limits instead of 'x' limits if you are going to stick with the variable z. Otherwise just find the indefinite integral in terms of z and change the function back to x.
 
Dick said:
http://en.wikipedia.org/wiki/Tangent_half-angle_formula You seem to have turned a '2' into a 'z' in you dx derivation. And you would have to change the limits to 'z' limits instead of 'x' limits if you are going to stick with the variable z. Otherwise just find the indefinite integral in terms of z and change the function back to x.

Okay, thanks for the help, I'm looking at the solution and it has 1 and \sqrt{3} for the bounds, but I do not get that when insert \frac{\pi}{3} and \frac{\pi}{2}.

EDIT:

I got it, guess I am blind. Thanks again.
 
Last edited:

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