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    Contour and Fresnel Integrals

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    We want to calculate the integral of the function x*sin(x^4), from 0 to infinity.

    We may think that this is similar to a Fresnel Integral (sin(x^2)). In that case, we would set z=e^(iz)^2, and then integrate over the special contour with regions I: 0 to R, II: theta = 0 to pi/4 and then finally to region III: R to 0 along e^i*pi/4. Now, this leads to the problem of integrating the function x*e^(i*x)^4.

    A much simpler approach is suggested.

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    https://brainmass.com/math/integrals/contour-fresnel-integrals-517554

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    The solution is the integral (with limits between 0 and infinity) of ...

    Solution Summary

    We calculate the integral of the function x*sin(x^4), from 0 to infinity using the known value at infinity of the Fresnel integral of sin(x^2).

    $2.49

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