Can anyone help me with these integral?

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  • 4 weeks ago
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    (2)  Think of the integral as the volume over the unit square of an object whose lower boundary is at z = 0 and whose upper boundary is at

    z = (x - y)*sin(xy)/(x^2 + y^2).  You might have to investigate whether the volume is finite, but it clearly does not matter whether the volume is evaluated by taking slices parallel to the xz-plane or slices parallel to yz-plane.  Slicing parallel to the xz-plane means doing the "dx" integration first; slicing parallel to the yz-plane means doing the "dy" integration first.

    (3)  I'll do the "dx" integration first.  The indefinite integral of

    xe^(-kx^2) dx 

    would best be attacked by letting u = kx^2, du = 2kx dx, so you'd have a new integrand of

    (1/(2k))e^(-u) du

    and the indefinite integral would be

    -(1/(2k))e^(-u) = -(1/(2k))e^(-kx^2).

    At x = infinity you get zero; at x = 0 you get -1/(2k), but it is to be subtracted, so it's 1/(2k).

    In your problem the "k" would be (1+y^2), so now you have

    the integral from y = 0 to infinity of

    [1/(2+2y^2)] dy.

    Note that the exponential disappeared entirely!

    Looks like you're going towards some answer with arctangent in it.  Shouldn't be too hard?

    (1)  I don't know.

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