Integration by trigonometric substitution

  • Thread starter cmajor47
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
Defennder
Homework Helper
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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?
 
  • #3
57
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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?
 
  • #4
Defennder
Homework Helper
2,591
5
Well as I hinted earlier, what is the derivative of csc^n theta?
 

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