Inverse Substitution: Solving for z in Terms of x

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Homework Help Overview

The discussion revolves around the process of making an inverse substitution in an integral, specifically substituting x = 2tan(z) and subsequently expressing z in terms of x after evaluating the integral.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore different methods to express z in terms of x after performing the substitution, including the use of right triangles and trigonometric identities.

Discussion Status

Participants are actively engaging with the problem, with some suggesting the use of a right triangle to derive relationships between the variables. There is a recognition of the challenge in expressing z without directly using arctan, leading to alternative approaches being discussed.

Contextual Notes

There is a specific constraint mentioned regarding the prohibition of stating the answer in terms of arctan, which influences the approaches being considered by participants.

jumbogala
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Homework Statement


Let's say you make the inverse substitution x = 2tan(z) in some integral.

Let's say you evaluate the integral get something like like 4sin(z). How do you put z back in terms of x?

Homework Equations





The Attempt at a Solution


I can do it by saying arctan(x/2) = z, but my teacher said we are not allowed to state the answer that way. I'm not sure how else you would do that...?
 
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Draw a right triangle where tan(x)=x/2. For example, let the side opposite the angle z be x and the side adjacent be 2. Now use Pythagoras to find the hypotenuse. What then is sin(z) in terms of x? It's opposite over hypotenuse, right?
 
Ohh okay, so in this case you would end up with 4x / sqrt(x^2 + 4).

Thanks!
 
Exactly. 4*sin(arctan(x/2))=4x/sqrt(x^2+4). Drawing a triangle is nice way to derive stuff like that without memorizing it.
 

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