Ahmed's Integral

Ahmed's integral is the definite integral

👁 int_0^1(tan^(-1)(sqrt(x^2+2)))/(sqrt(x^2+2)(x^2+1))dx=5/(96)pi^2

(OEIS A096615; Ahmed 2002; Borwein et al. 2004, pp. 17-20).

This is a special case of a general result that also yields

👁 int_0^1(tan^(-1)(sqrt(x^2+1)))/((x^2+1)^(3/2))dx =(1/4-1/2sqrt(2))pi+3/2sqrt(2)tan^(-1)(sqrt(2)) int_0^1(tan^(-1)x)/(x(x^2+1))dx=1/2K+1/8piln2

(OEIS A102521 and A098459) as additional cases (Borwein et al. 2004, p. 20), where 👁 K
is Catalan's constant.

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References

Ahmed, Z. "Definitely An Integral." Amer. Math. Monthly 109, 670-671, 2002.Borwein, J.; Bailey, D.; and Girgensohn, R. "Ahmed's Integral Problem." §1.6 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 17-20, 2004.Sloane, N. J. A. Sequences A096615, A098459, and A102521 in "The On-Line Encyclopedia of Integer Sequences."

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Ahmed's Integral

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Weisstein, Eric W. "Ahmed's Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AhmedsIntegral.html

URL: https://mathworld.wolfram.com/AhmedsIntegral.html

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Ahmed's Integral

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References

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