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Stuck on this integral (e^x)

  1. Mar 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Integrate

    -9e^x - 28 / e^2x + 9e^x + 14

    It gives a hint which is substitute u = e^x.


    2. Relevant equations



    3. The attempt at a solution
    I want to integrate by partial fractions if possible... however before I can do that, I need to make the substitution, and I can't figure out how.

    If I take u = e^x, then du=e^x dx .

    But I have no e^x dx by itself in my equation to replace?
     
  2. jcsd
  3. Mar 11, 2009 #2

    lanedance

    User Avatar
    Homework Helper

    note you can rewrite it as
    [tex]
    e^{-x}du = dx
    [/tex]

    so then
    [tex]
    \frac{du}{u}= dx
    [/tex]
     
  4. Mar 11, 2009 #3

    Mark44

    Staff: Mentor

    Here's what I think your integral is, with dx:
    [tex]\int \frac{-9e^x - 28}{e^{2x} + 9e^x +14}dx[/tex]

    Since you didn't use any parentheses, it's difficult to tell what the original problem really is, so I wrote the integral as what I thought you meant.

    If du = e^x dx, then dx = du/(e^x) = du/u.

    Make the substitution, and we'll take it from there.
     
    Last edited: Mar 11, 2009
  5. Mar 11, 2009 #4
    Hmm, okay.

    So that gives

    (-9u-28)/((u^2)+9u+14), that whole thing multiplied by du / u.

    Is that right?
     
  6. Mar 11, 2009 #5

    lanedance

    User Avatar
    Homework Helper

    sounds alright to me, try and factor the denominator as well
     
  7. Mar 11, 2009 #6
    The denominator factors out into
    (u + 2)(u + 7)(u).

    So from here I can use partial fractions to integrate, I think.

    I will have three terms to integrate, which I'll add together at the end:

    A / (u + 2)
    B / (u + 7)
    C / u

    I need to solve for A B and C then integrate. Is this the right apporach, or is there an easier way? Solving for the ABC seems complicated.
     
  8. Mar 11, 2009 #7

    Mark44

    Staff: Mentor

    Looks good, so far.
     
  9. Mar 11, 2009 #8
    Thanks for your help, both of you. I am not going to continue with the rest of the problem because I know how to solve it, and I still have some other practice questions to do.
     
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