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Integration by Trigonometric Substitution

  1. Aug 31, 2009 #1
    1. The problem statement, all variables and given/known data
    Evaluate:
    [tex]\int\frac{1}{(4 - \tan^2{x})^{3/2}}dx[/tex]


    2. Relevant equations
    I must integrate the above equation using only trigonometric subtitutions of algebraic equations.


    3. The attempt at a solution
    Here is what I have so far:

    [tex]Let \tan{(x)} = 2\sin{(\theta)}[/tex]

    [tex]x = \tan^{-1}{(2\sin{(\theta)})}[/tex]

    [tex]dx = \frac{4\sin{(\theta)}\cos{(\theta)}}{1 + 4\sin^2{(\theta)}}d\theta[/tex]

    [tex]\int\frac{1}{(4 - \tan^2{x})^{3/2}}dx = \int\frac{1}{(4 - (2\sin{(\theta)})^2)^{3/2}}\frac{4\sin{(\theta)}\cos{(\theta)}}{1 + 4\sin^2{(\theta)}}d\theta[/tex]

    [tex]= \frac{4}{8}\int\frac{\sin{(\theta)}\cos{(\theta)}}{(1 + 4\sin^2{(\theta)})(1 - \sin^2{\theta)})^{3/2}}d\theta[/tex]

    [tex]=\frac{1}{2}\int\frac{\sin{(\theta)}\cos{(\theta)}}{(1 + 4\sin^2{(\theta)})\cos^3{(\theta)}}d\theta[/tex]

    [tex]=\frac{1}{2}\int\frac{\sin{(\theta)}\sec^2{(\theta)}}{1 + 4\sin^2{(\theta)}}d\theta[/tex]

    [tex]=\frac{1}{2}\int\frac{\sec{(\theta)}\tan{(\theta)}}{1 + 4\sin^2{(\theta)}}d\theta[/tex]

    I can't seem to integrate the final integral above. Can anybody help me get past this step or can anybody tell me if I've made a mistake. Thanks!
    -James

    P.S. Sorry if it's messy!
     
  2. jcsd
  3. Aug 31, 2009 #2
    Sorry to reply to my own post but I've solved the problem and I do not know how to delete this. Thank you!
    -James
     
  4. Aug 31, 2009 #3

    Mark44

    Staff: Mentor

    Did the substitution you used work out? It doesn't look like it produced anything that would be useful.
     
  5. Aug 31, 2009 #4
    Your substitution tanx = 2sinΘ doesn't look valid at all. I would have tried x = tanΘ. What did you get for your answer?
     
  6. Aug 31, 2009 #5
    No actually I managed to waste a lot of time after I copied the problem down wrong from the book. I gues that doesn't change the fact that I still don't know how to integrate the problem I posted but it no longer matters.
     
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