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Homework Help: Hard Indefinite Integral

  1. Feb 20, 2010 #1
    1. The problem statement, all variables and given/known data[/b]

    [tex]\int e^{3x}\sqrt{1+e^{2x}[/tex]dx

    2. Relevant equations

    Parts of Integration

    3. The attempt at a solution

    Started off using U substitution setting ex = to u. Then tried to use parts of integration. Now I am stuck.
  2. jcsd
  3. Feb 20, 2010 #2
    This is a very messy problem.
    After u-substitution (let u = e^{x}), you should get
    [tex]\int u^2\sqrt{1+u^2}du[/tex]

    Then I imagine you should try u-substitution using trig functions like tan and sec. It gets very convoluted very quickly.
  4. Feb 20, 2010 #3
    Thats what I have done and it seems to continue to get larger. Just trying to make sure I was on the right track. Thanks for your help
  5. Feb 20, 2010 #4
    Instead of committing fully to one particular approach, you should do some exploratory computations to see what works best. That will often save you from persuing some tedious method if a very simple method is available. In this case you missed a partial integration step where you integrate the factor u sqrt(1+u^2) and thus have to evaluate the integral of (1+u^2)^(3/2), which suggests substituting u = sinh(t) leaving you having to integrate cosh^4(t), which is trivial.
  6. Feb 21, 2010 #5
    Haven't learned U=sinh(t). Only using u-sub, Trig sub, and parts. I think thats what is making it so long and messy. If any other suggestions it would be much appreciated.
  7. Feb 21, 2010 #6
    This requires hyperbolic sin.
  8. Feb 22, 2010 #7
    Haven't tried this out, just the first thing I thought of:
    e3x√(1 + e2x) = ex·e2x√(1 + e2x)

    You can try integration by parts, integrating the right side with the substitution u = 1 + e2x

    Whatever you do, you won't have to go into hyperbolic trig functions, even if the regular trig functions make the integration a little messy.
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