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Math Help - integration

  1. #1
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    integration

    how do you show that intergrate from o to infinity of exp (-r^2) over (r^4 + a^4) is less than pi over ( (2)^(0.5) a^3)

    given that integrate from o to infinity of 1/ (x^2 +a^2 ) dx is pi over (2a)
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    how do you show that intergrate from o to infinity of exp (-r^2) over (r^4 + a^4) is less than pi over ( (2)^(0.5) a^3)

    given that integrate from o to infinity of 1/ (x^2 +a^2 ) dx is pi over (2a)
    Since e^{-r^2}<1 whenever r>0, it follows that \int_0^\infty\!\!\frac{e^{-r^2}}{r^4+a^4}\,dr < \int_0^\infty\!\!\frac1{r^4+a^4}\,dr . Substitute r=ax into that second integral to see that it is equal to \frac1{a^3}\int_0^\infty\!\!\frac1{1+x^4}\,dx. The integral \int_0^\infty\!\!\frac1{1+x^4}\,dx can be evaluated (by contour integration, or by partial fractions), and it is equal to \frac{\pi}{2\sqrt2}.

    Putting that all together, you get \int_0^\infty\!\!\frac{e^{-r^2}}{r^4+a^4}\,dr < \frac{\pi}{2a^3\sqrt2}. That is better than what was asked for, because it has an extra 2 in the denominator. But I don't see any way of getting this result by using the given hint about \int_0^\infty\!\!\frac1{x^2+a^2}\,dx.
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