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Difficult integral.

  1. Nov 3, 2009 #1
    1. The problem statement, all variables and given/known data
    The whole integral is a double integral, but I can't even perform the first integration.

    The integrand is cosx*sqrt(3+(cosx)^2) dxdy
    The x bounds are from arctan(y) to pi/4
    the y bounds are from 0 to 1

    2. Relevant equations
    Converting rectangular to polar may help.

    3. The attempt at a solution

    I've tried parts, but it was a disaster. I also started converting it to polar, but that was also a disaster.
  2. jcsd
  3. Nov 3, 2009 #2
    I switched dx and dy and the bounds.

    The new y bounds are from 0 to tanx
    The new x bounds are from 0 to pi/4

    The resulting integrand after successfully integrating in terms of y is

    -sinx*sqrt(3 + (cosx)^2) dx

    It's a little nicer, but still ugly.

    Putting the integrand into the mathematica online integrator, I get this:

    (-3*ArcSinh[Cos[x]/Sqrt[3]])/2 - (Cos[x]*Sqrt[3 + Cos[x]^2])/2

    How do I get here? I tried by starting with parts, but it doesn't get me far.
  4. Nov 3, 2009 #3
    Solved with trig substitution and then parts.
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