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Watch your integral.

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  • Watch your integral.

    This puzzle is slightly more trickier than the one I posted a moment ago. Usually when teaching young people the subtle mistakes you can make with integrals, I usually give them this puzzle after they've learned the rule of doing integration by parts (especially if they think they know what they're doing).

    In highschool notation the rule for integration by parts is stated as



    Where



    and



    Now lets do the following integral



    this gives us



    Lets subtract the integral from both sides and we're now left with


  • #2
    I never noticed this and can't figure it out.

    I would first reduce the integrand to 1 and get x + C

    K54

    Comment


    • #3
      Originally posted by klaus54 View Post
      I never noticed this and can't figure it out.

      I would first reduce the integrand to 1 and get x + C

      K54
      P.S. You can express 1 some other ways and get the same "contradiction". E.g., try it with integrand X*X^-1, x=0.

      Comment


      • #4
        Originally posted by klaus54 View Post
        I never noticed this and can't figure it out.

        I would first reduce the integrand to 1 and get x + C

        K54
        Using integration by parts, as specified:

        f(x) = e^x
        g(x) = e^-x

        F(x) = e^x
        g'(x)= -e^-x

        So

        F(x)g(x) = 1

        But you're quite right to note the contradiction stems from the constant of integration.

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        • #5
          Originally posted by Leonhard View Post


          Lets subtract the integral from both sides and we're now left with

          Ooooh, I dig this one. I'm putting this in my pocket for later-- perhaps an extra credit assignment for students.
          "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
          --Thomas Bradwardine, De Continuo (c. 1325)

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          • #6
            I'm also kinda fond of it, can't remember where I've seen it though.

            Comment

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