Answer
$4 \ln 3-2 \ln 5$
Work Step by Step
Given: $ f(x, y, z)=\dfrac{x}{(y+z)^2}$
The iterated integral can be calculated as:
\begin{aligned}
\iiint_{\mathcal{B}} f(x,y,z)d V &= \iiint_{\mathcal{B}} \dfrac{x}{(y+z)^2} d V \\
&=\int_{2}^{4} \int_{-1}^{1} \int_{0}^{2} \dfrac{x}{(y+z)^2}dx d z d y \\
&= \int_{2}^{4} \int_{-1}^{1} [\dfrac{x^2}{2(y+z)^2}]_0^2 dzdy\\
&=[2 \ln |z-1|-2 \ln |1+z|]_2^4 \\
&=[2 \ln |4-1|-2 \ln |1+4|]-[2 \ln |2-1|-2 \ln |1+2|] \\ &=4 \ln 3-2 \ln 5
\end{aligned}
