1. **State the problem:** Calculate the definite integral $$5 \int_1^5 \frac{2x^2}{3} \, dx$$. 2. **Formula and rules:** The integral of a function $f(x)$ over $[a,b]$ is given by $$\int_a^b f(x) \, dx = F(b) - F(a)$$ where $F(x)$ is the antiderivative of $f(x)$. 3. **Rewrite the integral:** $$5 \int_1^5 \frac{2x^2}{3} \, dx = \frac{10}{3} \int_1^5 x^2 \, dx$$. 4. **Find the antiderivative:** $$\int x^2 \, dx = \frac{x^3}{3} + C$$. 5. **Evaluate the definite integral:** $$\frac{10}{3} \left[ \frac{x^3}{3} \right]_1^5 = \frac{10}{3} \left( \frac{5^3}{3} - \frac{1^3}{3} \right) = \frac{10}{3} \left( \frac{125}{3} - \frac{1}{3} \right) = \frac{10}{3} \times \frac{124}{3} = \frac{1240}{9}$$. 6. **Final answer:** $$\boxed{\frac{1240}{9}}$$.