Multiple Integrals

Calculus Level 5

\[\large \displaystyle \iiint_{V} \dfrac{x^2 y^2 z^2}{(x+y+z)^9}\ln^2 (1-x-y-z) \, dx \; dy \; dz \]

Evaluate the triple integral above where \(V: \{x,y,z\; | \; x,y,z>0,x+y+z\le 1 \}\).

If this integral can be expressed as \( \dfrac{\zeta( P)}Q \), where \(P \) and \(Q\) are positive integers, submit your answer as \(\dfrac QP \).

\[\] Notation: \(\zeta(\cdot) \) denotes the Riemann zeta function.

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