Mastering Trig Substitution for Evaluating Integrals of 1/x(4x+1)^1/2 in Calc II

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Homework Help Overview

The discussion revolves around evaluating the integral of 1/x(4x+1)^(1/2), with a focus on potential substitution methods, particularly trigonometric substitution and algebraic manipulation.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster considers using trigonometric substitution, specifically suggesting x=sec(u) for the expression 4x+1. However, uncertainty about the next steps is expressed. Another participant proposes an alternative approach using u^2=4x+1, indicating that this could simplify the integral into a rational function suitable for partial fractions. Further discussion includes the suggestion of letting u equal the radical itself as a common technique.

Discussion Status

The conversation is active, with participants exploring different substitution strategies. While one participant suggests a trigonometric approach, another offers a more straightforward algebraic method. There is no explicit consensus, but various lines of reasoning are being examined, indicating a productive exchange of ideas.

Contextual Notes

Participants are navigating the complexities of integral evaluation, particularly in the context of calculus homework, where specific substitution methods may be preferred or required. The discussion reflects varying levels of familiarity with integration techniques.

GravityGirl
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evaluate the integral of 1/x(4x+1)^1/2

so what i am thinking is that i will have to use a trig substution some how for
4x+1

maybe x=secu since the substution for a^2+x^2=sec^2u

but i am not sure how to go ahead and do it

please help me
 
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No. Don't do trig. Try u^2=4*x+1. You'll get a simple rational function you can then do with partial fractions.
 
GravityGirl said:
evaluate the integral of 1/x(4x+1)^1/2

so what i am thinking is that i will have to use a trig substution some how for
4x+1

maybe x=secu since the substution for a^2+x^2=sec^2u

but i am not sure how to go ahead and do it

please help me
Well, as Dick pointed out, you should let [tex]u = \sqrt{4x + 1}[/tex]
Or, in general, if you meet some radical, then it's common to let u = that whole radical, and then go from there.

You only let x = sec(u), when you meet: [tex]\sqrt{x ^ 2 - 1}[/tex], or when encountering [tex]\sqrt{x ^ 2 - a ^ 2}[/tex], you then let x = a sec(u).

Can you go from here? :)
 
Last edited:
I guess if she really wanted to, she could have done x= 1/4 tan^2 u. Takes much much longer though.
 

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