# 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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