How do I solve an indefinite integral with a trigonometric substitution?

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Homework Help Overview

The discussion revolves around solving an indefinite integral involving a trigonometric substitution, specifically the integral of 1/[(x^4)(sqrt(9x^2 -1))]. Participants are exploring the steps necessary to simplify and evaluate this integral using trigonometric identities and substitutions.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to use the substitution x=3secy and dx=3secytanydy, but expresses uncertainty about the correctness of this approach and how to proceed further. Other participants discuss the implications of the substitution and the resulting integral form, questioning the steps taken and suggesting further simplifications.

Discussion Status

Participants are actively engaging with the problem, with some providing guidance on the next steps after the substitution. There is a collaborative effort to clarify the integral's transformation and to explore the implications of the trigonometric identities involved. Multiple interpretations of the integral's evaluation are being discussed, but no explicit consensus has been reached.

Contextual Notes

There is a focus on ensuring that the substitutions and transformations align with trigonometric identities, and participants are considering the implications of these transformations on the integral's evaluation. The original poster's request for step-by-step help indicates a desire for thorough understanding rather than a straightforward solution.

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1. indefinite integral of 1/[(x^4)(sqrt(9x^2 -1))]

I'm looking for step by step help here. I use x=3secy and dx=3secytanydy to convert the top function to integral of: [3secytanydy]/[(tany)(3secy)^4]

I am not sure if this is right/how to get further
 
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mat331760298 said:
1. indefinite integral of 1/[(x^4)(sqrt(9x^2 -1))]

I'm looking for step by step help here. I use x=3secy and dx=3secytanydy to convert the top function to integral of: [3secytanydy]/[(tany)(3secy)^4]

I am not sure if this is right/how to get further

I labelled by triangle with hypotenuse = 3x, adjacent side = 1, opp. side = sqrt(9x^2 - 1), with angle t.

So sec t = 3x ==> x = 1/3 sec t
dx = 1/3 sec t * tan t * dt
sqrt(9x^2 -1) = tan t

Then,
\int \frac{dx}{x^4 \sqrt{9x^2 - 1}} = \int \frac{1/3 sec(t)tan(t)dt}{(1/3)^4 sec^4(t)~tan(t)}
Can you continue from there?
 


do you get integral of:27*(cost)^3 dt ?
 


then you integrate to get 27(sint-(1/3)(sint)^3) and use opp side over hypotenuse to replace with sint?
 


Yes. After you get the antiderivative, undo your substitution, and you'll be done.
 

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