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Integral Trouble

  1. Aug 27, 2005 #1
    [tex]\int ln(2x+1)dx[/tex]

    So far I know that I need to use integration by parts, I let [itex]u= ln(2x+1) [/itex] and so [itex] du= \frac {dx}{2x+1} [/itex]. Also, I said [itex] dv= dx [/itex] and [itex] v=x [/itex].

    So then plugging this into the equation for integration I get:

    [tex] xln(2x+1) - \int \frac {2x}{2x+1}dx [/tex]

    Then I determine that I need to do integration by parts again on the latter half of the function. So, for [itex]\int \frac {2x}{2x+1}dx [/itex], I let [itex]u= 2x [/itex] and so [itex] du= xdx [/itex]. Also, I said [itex] dv= \frac {dx}{2x+1} [/itex] and [itex] v= \frac {ln(2x+1)}{2} [/itex].

    So then plugging this into the equation for integration I get:

    [tex]\int \frac {2x}{2x+1}dx = xln(2x+1) - \int ln(2x+1)dx [/tex]

    Now, I have like terms so I say that:

    [tex]\int ln(2x+1)dx = xln(2x+1) - [xln(2x+1) - \int ln(2x+1)dx]. [/tex]

    I am not sure where I made an error here. Any help is appreciated.
     
  2. jcsd
  3. Aug 27, 2005 #2

    Galileo

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    Oi, so complicated >_<

    You don't have to integrate by parts again. You can split the fraction like:

    [tex]\frac{2x}{2x+1}=1-\frac{1}{2x+1}[/tex]

    But even simpler is first solving:

    [tex]\int \ln x dx[/tex]
    then the integral should be a piece of cake. The straight way to Rome is not always the shortest, nor the easiest to follow.
     
  4. Aug 27, 2005 #3
    [tex]\int ln(2x+1)dx = xln(2x+1) - \int \frac {2x}{2x+1}dx = xln(2x+1) - \int 1 - \frac {1}{2x+1} dx[/tex]
     
  5. Aug 27, 2005 #4
    Wow, how did I miss that, thanks so much.
     
  6. Aug 27, 2005 #5

    Hurkyl

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    Anyways, as to the actual method you implemented, your second integration was simply undoing the first integration by parts you tried. That's why you got a useless result at the end.
     
  7. Aug 27, 2005 #6

    Hurkyl

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    It's not productive to do the problem for the person asking for help. (especially when you do it wrong!) Fortunately, the poster had already figured it out from the hints!
     
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