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How to calculate the integral with the condition in wolfram

How to calculate the integral with the condition in Wolfram like

Integral(1/(x^4 + y^4)dxdy) where y >=x^2+1

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Solution

  • I assume you mean the definite integral over all x, y>x^2+1. The syntax is this:

    Integrate[1/(x^4 + y^4), {x, -Infinity, Infinity},{ y, x^2 + 1,Infinity}]

    Note mathematica's ordering of the integration variables is reverse of standard convention, ie. left to right is outside to inside. This takes quite a while to report that it does not converge. However the numerical integration gives a result:

    NIntegrate[1/(x^4 + y^4), {x, -Infinity, Infinity},{ y, x^2 + 1,Infinity}]

    0.389712

    My guess is the numeric result is correct and mathematica is simply wrong about the analytic convergence. You might try math.stackexchange.com or mathematica.stackexchange.com if you need to prove convergence. I am doubtful there is a nice analytic result.