Tuesday, 19 August 2014

integration - How do I Evaluate the Integral intinftyinftyfrac1ex2+1,dx?



Like the question states, how do I evaluate the integral
1ex2+1dx


I know of no methods to evaluate this, but when I plug this into Mathematica and Wolfram Alpha, it returns
1ex2+1dx=(12)πζ(12)

where ζ(x) is the Riemann Zeta Function.


Answer



The given integral equals

2+0dx1+ex2=+0dzz(1+ez)=n1(1)n+1+0enzzdz


or
πn1(1)n+1n=πη(12)=(12)πζ(12)

as claimed by WA.


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