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Integration by trigonometric substitution

  1. Oct 14, 2008 #1
    1. The problem statement, all variables and given/known data
    Solve the integral using trigonometric substitution.
    [tex]\int[/tex][tex]\frac{\sqrt{4x^{2}+9}}{x^{4}}[/tex]dx


    2. Relevant 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]

    3. 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.
     
  2. jcsd
  3. Oct 15, 2008 #2

    Defennder

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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?
     
  4. Oct 15, 2008 #3
    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?
     
  5. Oct 15, 2008 #4

    Defennder

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    Well as I hinted earlier, what is the derivative of csc^n theta?
     
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