Integration by trigonometric substitution

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Homework Help Overview

The discussion revolves around solving an integral using trigonometric substitution, specifically the integral of the form \(\int\frac{\sqrt{4x^{2}+9}}{x^{4}}dx\). The subject area is calculus, focusing on integration techniques.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to apply trigonometric substitution and has transformed the integral into a new form involving trigonometric functions. Some participants question the accuracy of the transformation and the resulting integrals, while others suggest exploring the properties of trigonometric derivatives to facilitate the integration process.

Discussion Status

The discussion is ongoing, with participants actively correcting each other's work and exploring different approaches to simplify the integral. There is a focus on identifying errors and clarifying the steps needed to progress towards a solution.

Contextual Notes

Participants are working within the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. There is an emphasis on understanding the differentiation and integration of trigonometric functions.

cmajor47
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Homework Statement


Solve the integral using trigonometric substitution.
[tex]\int[/tex][tex]\frac{\sqrt{4x^{2}+9}}{x^{4}}[/tex]dx


Homework Equations


2x=3tan[tex]\theta[/tex]
x=2/3 tan[tex]\theta[/tex]
dx=2/3 sec2[tex]\theta[/tex]d[tex]\theta[/tex]
[tex]\sqrt{4x^{2}+9}[/tex]=3sec[tex]\theta[/tex]

The Attempt at a Solution


[tex]\frac{8}{9}[/tex] [tex]\int[/tex] [tex]\frac{cos^{2}\theta}{sin^{4}\theta}[/tex] d[tex]\theta[/tex]

From here, I don't know how to get the integral into a form that is easy to solve.
 
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You made an error. Your numerator should be cos theta, not cos^2 theta. Now what do you know about how to differentiate and integrate powers of csc theta?
 
I fixed the error and have gotten to
[tex]\int[/tex]cot[tex]\theta[/tex]csc3[tex]\theta[/tex]d[tex]\theta[/tex]
But how do I get this to any easy to solve integral?
 
Well as I hinted earlier, what is the derivative of csc^n theta?
 

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