Find the mass of a rod of length 2m with density $\delta$(x) = $\sqrt{2x}$ kg/m given at a distance of x m from the left end. I have that mass is equal to $\int_0^2\delta(x)dx$, which when I substitute in the value for $\delta(x)$, I get $8/3$ kg, but the solutions for this problem show that the answer is $2\sqrt6$ kg. Am I missing something?

Your correct is correct for the linear density function $\delta(x)$ that was specified.

Write $\delta(x)=2^{1/2}x^{1/2}$ so that it is easier to apply the power rule in reverse when integrating:
$$m=\int_{0}^{2}{\delta(x)\,dx}=2^{1/2}\int_{0}^{2}{x^{1/2}\,dx}=2^{1/2}\left[\tfrac{2}{3}x^{3/2}\right]_{0}^{2}=2^{1/2}\times\tfrac{2}{3}\times(2^{3/2}-0)=\frac{8}{3}\text{ kg}$$
