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Integral from 0 to infinity of dx/(x^4 + x^2 + 1)

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Evaluate the Integral from 0 to infinity of dx/(x^4 + x^2 + 1).

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Solution Summary

A detailed solution is given regarding the integral from 0 to infinity.

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We can compute the integral:

K = Integral from 0 to infinity of dx/(x^4 + x^2 + 1)

as follows. The integrand is an even function of x, so we have:

2 K = Integral from minus infinity to infinity of dx/(x^4 + x^2 + 1)

We can evaluate this by considering the contour integral of dz/(z^4 + z^2 + 1) over the contour that starts at -R on the real axis and moves to R along the real axis and then we take a counterclockwise half circle with center the origin and radius R back to minus R. The contour will then enclose the poles in the upper half ...

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