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Integral of a vector valued function

  1. Feb 13, 2012 #1
    I came across this integral of a vector valued function.
    [tex] \int \mathbf A(t) \vec{w(t)} dt = \int \mathbf B(t) [/tex].
    I want to isolate [itex] \vec{w(t)} [/itex] and so I multiply by [itex] \left (\int \mathbf A(t) dt \right)^{-1} [/itex] on both sides.
    [tex] \left (\int \mathbf A(t) dt \right)^{-1} \int \mathbf A(t) \vec{w(t)} dt = \left (\int \mathbf A(t) dt\right)^{-1} \int \mathbf B(t) dt [/tex]

    I thought the correct form would be
    [tex] \int \vec{w(t)} dt = \left (\int \mathbf A(t) dt\right)^{-1} \int \mathbf B(t) dt [/tex].

    But it turns out I get the right answer if I take
    [tex] \vec{w(t)} = \left (\int \mathbf A(t) dt \right)^{-1} \int \mathbf B(t) dt [/tex].

    Can anyone show why the second form is correct?
    Last edited: Feb 13, 2012
  2. jcsd
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