Integration by Parts Contradiction

  1. Ok guys, this is my first post. Please go easy...:redface:

    This question is from Morris Kline's Calculus: An Intuitive and Physical Approach and unfortunately there aren't solutions for all questions (really annoying).

    I'm not even sure if this counts as a contradiction but anyway:

    Let us evaluate int.(1/x)dx by parts. If we let u=1/x and dv=1dx, we obtain int.(dx/x)=1 + int.(dx/x). Then 1=0. What is wrong?

    I would really appreciate a simple explanation from any of you experienced brains out there! Thanks.
     
  2. jcsd
  3. VietDao29

    VietDao29 1,422
    Homework Helper

    Here's what I think. Let:
    [tex]g(x) := \int f(x)dx \quad \mbox{and} \quad h(x):= \int f(x)dx[/tex]. Then, you do not have g(x) - h(x) = 0, you will have g(x) - h(x) = C, where C is some constant.
    So here's the same, you can say that:
    [tex]\int \frac{dx}{x} - \int \frac{dx}{x} = C[/tex], where C is some constant.
    So it's not a contradiction...
    Viet Dao,
     
  4. I get you Viet Dao... I don't think I would not have thought that way at all on my own... not tonight anyway. Thanks.

    Here's a thought I just had:
    Could int.(dx/x) = C + int.(dx/x) , where C is some constant other than 1, be eventuated from Int. by Parts?
     
  5. VietDao29

    VietDao29 1,422
    Homework Helper

    You can continue integrating by parts, something like:
    [tex]\int \frac{dx}{x} = 1 + \int \frac{dx}{x} = 1 + \left( 1 + \int \frac{dx}{x} \right) = 1 + 1 + .. + 1 + \left( 1 + \int \frac{dx}{x} \right)[/tex].
    So you'll have:
    [tex]\Leftrightarrow \int \frac{dx}{x} - \int \frac{dx}{x} = 1 + 1 + 1 + ... + 1[/tex].
    Viet Dao,
     
  6. Gotcha, excellent explanation.
     
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