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A simple question about integration by substitution

  1. Oct 9, 2011 #1
    Hello all,

    We've just begun integration in my maths class and I have a question about a certain aspect of integration by substitution.

    Let's say for instance you let u = 2x-1. Then you differentiate it and get du/dx = 2.

    My maths teacher said " you can now think of it as multiplying across by 'dx' ".

    Which leads to du = 2dx.

    My question is, what mathematical operations are you actually committing to achieve this line?

    Thanks in advance for any help.
  2. jcsd
  3. Oct 9, 2011 #2


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    It is a symbolic multiplication, since du and dx are symbols not numbers.
  4. Oct 9, 2011 #3


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    If your integration were, say,
    [tex]\int e^{2x- 1}dx[/tex]
    you can think in either of two ways:
    If u= 2x- 1 then du= 2dx so dx= (1/2)du and the integral becomes
    [tex]\int e^{2x- 1}dx= \int e^u((1/2)du)= (1/2)\int e^u du[/tex]

    Or if u= 2x- 1 then du 2dx and if we multiply and divide by 2 we get
    [tex]\int e^{2x-1}dx= (1/2)(2)\int e^{2x-1}dx= (1/2)\int e^{2x-1}(2dx)= (1/2)\int e^u du[/tex]

    Notice that in the first case we are moving the "1/2" outside the integral and in the second case, we are moving the "2" inside the integral. We can do that because the derivative of 2x is a constant. If the integral were
    [tex]\int e^{x^2}dx[/tex]
    so that taking [itex]u= x^2[/itex] gives [itex]du= 2xdx[/itex], we cannot do that substitution. In fact, that integral is not any "elementary" function.
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