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Integral of xcos(xy) over a rectangle

  1. May 1, 2012 #1
    Hi there. I'm having some trouble with some double integrals here. All of them are to be evaluated on the rectangle[itex]_{}[/itex] [itex]1≤x≤2, 0≤y≤1[/itex], and the functions are:

    1: [itex]\int\int_{A}\frac{1}{x+y}dxdy[/itex].
    On this one I made [itex]α(y)=\int^{2}_{1}\frac{1}{x+y}dx=ln(\frac{2+y}{1+y})[/itex], and finally I should evaluate [itex]\int^{1}_{0}α(y)dy[/itex], and this is where I got stuck.

    2: [itex]\int\int_{A}xcos(xy)dxdy[/itex].
    I again have evaluated [itex]α(y)[/itex], using integration by parts, but then I got stuck with some integrals like [itex]\int\frac{sin(y)}{y}dy[/itex], [itex]\int\frac{cos(2y)}{y^{2}}dy[/itex], and so on, which I couldn't find the primitive.

    Any tips on how to evaluate those?
  2. jcsd
  3. May 1, 2012 #2


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    You are fully allowed, by Fubini's theorem, to switch the order of integration.
  4. May 1, 2012 #3
    That's it, for both cases. Many thanks.
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