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Integration: what variables can you move outside of the integrand?

  1. Jan 24, 2013 #1
    [tex]1.\int { x } dx=x\int { 1 } dx\\ 2.\int { t } dx=t\int { 1 } dx\\ 3.\int _{ x }^{ x+1 }{ x } dt=x\int _{ x }^{ x+1 }{ 1 } dt[/tex]

    Which of the equations are correct?
     
  2. jcsd
  3. Jan 24, 2013 #2

    Zondrina

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    3 and 2 are both correct.
     
  4. Jan 24, 2013 #3

    Mark44

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    And 1 is incorrect. The following is a property of integrals:
    ##\int k~f(x)~dx = k\int f(x)~dx##, for k a constant, but there is no property that says you can move a variable across the integral sign.
     
  5. Jan 24, 2013 #4

    Zondrina

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    Integral xdx is certainly not the same as x times integral dx.
     
  6. Jan 25, 2013 #5

    HallsofIvy

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    You can move constants (and so variables that are independent of the variable of integration and so are treated like constants in the integration) outside the integral.

    No, x is the variable of integration so we cannot take it outside the integral.
    The integral on the left is [itex]x^2/2+ C[/itex] and on the right [itex]x(x+ c)= x^2+ cx[/itex].

    If we know that t is independent of x, then both integrals are "tx+ C". If t is a function of x then the first is still "tx+ C" but the other depends upon exactly what function of x t is.

    If x is independent of the variable of integration, t, both of those are the same and are equal to x(x+1- x)= x. If x is a function of t, then the left depends upon exactly what function of t x is while the right is still x.

     
  7. Jan 26, 2013 #6
    Thanks for the clear-up. :)
     
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