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Math Help - A step in an integral question multiplying logs and fractions

  1. #1
    dix
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    A step in an integral question multiplying logs and fractions

    I have a problem understanding a part of a solution to a calculus question. It's a 'by-parts' question and it's close to the solution.

    = [ln2 (2^4/4)-(2^5/16)] - [-1/16]

    I don't understand this next step:
    =4ln2 (2^4/4(4)) - (2^5/16) + (1/16)

    I understand that this part in red is being multiplied by 4 in order to get 16s in the denominators, but understand is how it's being multiplied by 4. Are the log and fraction together? How does (2^4/4)x4 become (2^4/4(4))?

    Thanks for any help
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  2. #2
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    e^(i*pi)'s Avatar
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    As it's written the fraction is not an argument of the log although for an integral output it seems very strange, what is the original question?

    What you've wrote is \ln(2) \times \dfrac{2^4}{4} - \dfrac{2^5}{16} - \left(-\dfrac{1}{16}\right) and when it reduces you get 4\ln(2) - 2 + \dfrac{1}{16}

    All I can see what they've done in red is multiply that first term by \dfrac{4}{4} = 1


    Could you post the original question so we can verify your solution?
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  3. #3
    dix
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    Thnaks. Here is the question and answer:

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  4. #4
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    e^(i*pi)'s Avatar
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    The solution to this integral is [\dfrac{1}{16}x^4 \times (4\ln(x) - 1)]^2_1 (I've tidied the syntax up a bit - post back if you didn't get this answer)

    [\dfrac{1}{16}(2^4) \times (4\ln(2)-1)] - [\dfrac{1}{16} \times -1] = 4\ln(2)-1 + \dfrac{1}{16} = \ln(16) - \dfrac{15}{16}

    I'm not sure how where you got the power of 5 from in your integral result from
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  5. #5
    dix
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    Thanks again. You're right about the 5 in the integral. I've uploaded a pic of what I went through and highlighted the part I don't understand. The multiplying to get the 16 on the bottom just seems arbitrary and I can't make sense of it. http://i.imgur.com/vDIGn.jpg
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  6. #6
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    e^(i*pi)'s Avatar
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    For what it's worth the you have the letters the 'wrong' way round (according to convention). Your calculations are fine though

    What's been done in the orange box is that they've multiplied top and bottom by 4, it's by no means an essential step and I'm not sure why they bothered to do it since it doesn't make a difference.

    edit: it looks like they've done it to give a common denominator of 16 across the whole expression. Since that's been done we can factor out 1/16 from the fraction to give \dfrac{1}{16} \left(2^4 \cdot 4 \ln(2) - 2^4 + 1\right) = \dfrac{1}{16}(64\ln(2) - 16 + 1)


    Looking at that first term (the one in orange) it is written as \ln(2) \cdot \dfrac{2^4}{4}. Since 2^4 = 16 and 16/4 = 4 we can cancel it down to \ln(2) \cdot 4 = 4\ln(2). In turn this equals ln(16) from the log power law.


    If we were to multiply top and bottom by 4 (as they did) we have : \ln(2) \cdot \dfrac{2^4}{4} \cdot \dfrac{4}{4} = 4 \cdot \ln(2) \cdot \dfrac{2^4}{4 \cdot 4} = 4\ln(2) \cdot \dfrac{16}{16} = 4\ln(2)




    Does that make sense (I'm not too sharp at explaining)
    Last edited by e^(i*pi); May 30th 2011 at 12:45 PM. Reason: white space+ see post (bold)
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  7. #7
    dix
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    Ah right yes, I can see it now. Thanks very much!
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