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Did I do this integral right?

  1. Feb 15, 2005 #1
    [tex]\int \frac {e^x+4}{e^x}dx =?[/tex]

    Here's what I did:
    [tex]\int \frac {e^x+4}{e^x}dx = \int e^{-x}(e^x+4)dx [/tex]
    [tex]\int e^{-x}(e^x+4)dx =\int 1+4e^{-x} = x-\frac {4e^{-x+1}}{x+1}[/tex]

    Did I do this correctly? Is there a more simplified answer?
     
  2. jcsd
  3. Feb 15, 2005 #2

    Galileo

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    Rewriting [itex]\frac{e^x+4}{e^x}=1+4e^{-x}[/itex] was correct.

    Check the antiderivative of [itex]e^{-x}[/itex]. Your answer is not correct. You can easily check it by differentiating it.

    Mind the difference between [itex]x^a[/itex] where the base is the variable and [itex]a^x[/itex] where the base is constant and the exponent is the variable.
     
  4. Feb 15, 2005 #3
    since [tex]\int e^{x} = e^{x}+C[/tex]
    then...
    [tex]\int 1+4e^{-x} = x+4e^{-x}+C[/tex]

    is that correct?
     
  5. Feb 15, 2005 #4
    When you differentiate [tex]x + 4e^{-x} + C[/tex] you get

    [tex] 1 - 4e^{-x} [/tex] , so the integral is actually

    [tex] \int 1 + 4e^{-x} dx = x - 4e^{-x} + C [/tex]
     
  6. Feb 15, 2005 #5
    I dont understand where the negative came from
     
  7. Feb 15, 2005 #6
    The integral of [tex] e^x dx = {e^x} + C[/tex]

    The integral of [tex] e^{-x}dx = -e^{-x} + C[/tex].

    Differentiating that answer you find that [tex] \frac {d}{dx} -e ^{-x} = e^{-x}
    [/tex]
     
  8. Feb 15, 2005 #7

    dextercioby

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    Think of it as an [itex] e^{u} [/itex] and apply the method of substitution:
    [tex] -x=u [/tex]

    Daniel.

    P.S.That's how u end up with the minus.
     
  9. Feb 15, 2005 #8
    [tex]\int \frac {e^x}{e^x+4}dx =?[/tex]

    Here's what I did:
    [tex]= \int e^{x}(e^x+4)^{-1}dx [/tex]
    subsitute:
    [tex]u=e^x+4[/tex]
    [tex]du=e^x dx[/tex]
    [tex]\int u^{-1}du =ln(e^x+4)[/tex]

    Did I do this correctly? Is there a more simplified answer?
     
  10. Feb 15, 2005 #9

    NateTG

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    Looks good to me.
     
  11. Feb 15, 2005 #10

    dextercioby

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    Don't forget the constant of integration.

    Daniel.
     
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