Answer
$$8 \ln 2$$
Work Step by Step
Our aim is to integrate the integral as follows:
$$\int^7_0 \int^2_0 \int^{\sqrt{4-q^2}}_0 \dfrac{q}{r+1} \space dp \space dq \space dr=\int^7_0 \int^2_0 \dfrac{q\sqrt{4-q^2}}{r+1} \space dq \space dr \\=\int^7_0 \dfrac{1}{3(r+1)}[-(4-q^2)^{3/2}]^2_0 dr\\=\dfrac{8}{3} \times \int^7_0 \dfrac{1}{r+1}dr \\=\dfrac{8 \ln 8}{3}\\=8 \ln 2$$
