Difficult integral for Trig Substitution

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Discussion Overview

The discussion revolves around the evaluation of the integral \(\int x^2\sqrt{(x^2-4)} dx\) using trigonometric substitution. Participants explore various substitution methods and express frustration over the complexity of the derivation and evaluation process.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses frustration with the integral and seeks clarity on the derivation process, noting that previous attempts at integration by parts led to more complicated integrals.
  • Another participant suggests using the substitution \(u = 2\sec(\theta)\) to simplify the integral.
  • A different participant reiterates the substitution \(x = 2\sec(\theta)\) and discusses the relationship between trigonometric identities, indicating that this substitution reduces the square root to \(2\tan(\theta)\).
  • One participant proposes an alternative substitution using hyperbolic functions, specifically \(x = 2\cosh(u)\).
  • Participants express confusion over the resulting integral \(\int \tan^2\theta \sec^3\theta d\theta\) and seek assistance in solving it.
  • Another participant points out a mistake regarding the derivative \(dx/d\theta\), indicating that it was overlooked in previous calculations.
  • One participant acknowledges a mistake in their calculations but believes it does not affect the integral's outcome, simplifying it to \(8\int \frac{\cos^5\theta}{\sin^3\theta} d\theta\) and continues to seek solutions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the integral, and multiple competing views regarding substitution methods remain. The discussion reflects uncertainty and ongoing exploration of the problem.

Contextual Notes

Participants mention various substitutions and transformations, but there are unresolved mathematical steps and assumptions that could affect the evaluation process. The complexity of the integral and the derivation remains a point of contention.

bigred09
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ok i have been studying the in-depth processes of trigonometric substitution with integrals and this problem has me frusterated.

[tex]\int x^2\sqrt{(x^2-4)} dx[/tex]

The evaluation is clear (from an old Table of Integrals I found), but the derivation is not at all clear, which is what i want to know.

I also tried to solve this by integration by parts, but every approach ended with an even more complicated integral, so trig substitution is probably the best choice.

Can anyone help?
 
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Try the substitution u = 2sec(theta)
 
bigred09 said:
ok i have been studying the in-depth processes of trigonometric substitution with integrals and this problem has me frusterated.

[tex]\int x^2\sqrt{(x^2-4)} dx[/tex]
[itex]sin^2(\theta)+ cos^2(\theta)= 1[/itex] so [itex]sin^2(\theta)= 1- cos^2(\theta)[/itex] and, dividing on both sides by [itex]cos^2(\theta)[/itex], [itex]tan^2(\theta)= sec^2(\theta)- 1[/itex]. The substitution [itex]x= 2sec(\theta)[/itex], as bigred09 suggested, will reduce that squareroot to [itex]2 tan(\theta)[/itex].

The evaluation is clear (from an old Table of Integrals I found), but the derivation is not at all clear, which is what i want to know.

I also tried to solve this by integration by parts, but every approach ended with an even more complicated integral, so trig substitution is probably the best choice.

Can anyone help?
 
Alternatively, use the hyperbolic substitution [tex]x=2Cosh(u)[/tex]
 
ok well with trig substitution, i get

[tex]\int tan^2\theta sec^3\theta d\theta[/tex]

which doesn't help me. Can someone solve this integral then?
 
bigred09 said:
ok well with trig substitution, i get

[tex]\int tan^2\theta sec^3\theta d\theta[/tex]

which doesn't help me. Can someone solve this integral then?

Wrong! Look what Halls said, post #3.
 
Letting

[tex]x=2sec(\theta)=>4sec^2(\theta)\sqrt{4(sec^2(\theta)-1)}=4sec^2(\theta)*2\sqrt{tan^2(\theta)}=...{[/tex]
Edit: Disregard this!
 
Last edited:
@ sutupidmath:

You're forgetting about dx/d(theta)
 
JG89 said:
@ sutupidmath:

You're forgetting about dx/d(theta)

:redface:
 
  • #10
right i actually forgot the coefficient 8 but that doesn't mess with the integral. and [tex]dx=sec\theta tan\theta[/tex]


so what halls said wass valid. all i did was simplify it more. even more so it looks like this:

[tex]8\int \frac{cos^5\theta}{sin^3\theta} d\theta[/tex]


so uh...seriously...any ideas on solving this?
 

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