Answer
$\dfrac{13 \delta}{6}$
Work Step by Step
We have $$m_x=\delta \int_0^2 ( x^{1/2}) \times \sqrt {1+\dfrac{1}{4x}} dx \\=\delta \int_0^2 (x+\dfrac{1}{4})^{1/2}dx$$
Set $\space (x+\dfrac{1}{4})=t$
Now, $$m_x=\delta \int_{1/4}^{9/4} t^{1/2} dt \\= \dfrac{2 \delta} {3}(\dfrac{27}{8}-\dfrac{1}{8} \\=\dfrac{13 \delta}{6}$$
