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Evaluating Indifinite Integral

  1. Dec 8, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]
    \int \sec^{2}(3x)e^{\tan(3x)}dx
    [/tex]

    2. Relevant equations

    The method I am trying to use is integration by substitution(u substitution).



    3. The attempt at a solution

    I start out by making u = tan(3x)

    So i end up having [tex]
    \int \sec^{2}(3x)e^{u}dx
    [/tex]

    Stuck after here though. How do I simplify this further?


    The final solution as per my professors solution sheet should be [tex]
    1/3e^{tan(3x)} + C
    [/tex]

    However, I cannot seem to figure out why this is the solution.


    -Thank you
     
  2. jcsd
  3. Dec 8, 2009 #2
    change sec^{2} 3x into 1 + tan^{2}

    Then make substitutions
     
  4. Dec 8, 2009 #3

    Dick

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    Homework Helper

    If u=tan(3x), then what's du?
     
  5. Dec 8, 2009 #4

    Mark44

    Staff: Mentor

    This is not a helpful suggestion.
     
  6. Dec 8, 2009 #5
    Would du be [tex]sec^{2}(3x) (3dx)[/tex]?
     
  7. Dec 8, 2009 #6

    Dick

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    Yes, it would. Can you use that to finish? Solve that for dx and put it into the original integral.
     
  8. Dec 8, 2009 #7
    hi, sorry for the late reply.


    Solving for dx I got

    dx= du/(Sec^2(3x)) x 3

    So

    Sec^2(3x)e^u du/sec^2(3x) x 3


    sec^2(3x) cancel each other out?

    so I'm left with

    e^u(du)(1/3) = 1/3e^u(du)

    so 1/3e^(tan(3x))+C?

    Is my approach correct?

    -Thank You
     
  9. Dec 8, 2009 #8

    Dick

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    Homework Helper

    Yes, yes, yes.
     
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