We are tasked with evaluating the definite integral:

$\int_{0}^{5} \left(3x^2 + 3\right) dx$

Step-by-step solution:

1. Split the integral: $\int_{0}^{5} \left(3x^2 + 3\right) dx = \int_{0}^{5} 3x^2 dx + \int_{0}^{5} 3 dx$

2. Evaluate the first integral: $\int 3x^2 dx = 3 \cdot \frac{x^3}{3} = x^3$ Applying the limits from 0 to 5: $\left[x^3\right]_0^5 = 5^3 - 0^3 = 125$

3. Evaluate the second integral: $\int 3 dx = 3x$ Applying the limits from 0 to 5: $\left[3x\right]_0^5 = 3(5) - 3(0) = 15$

4. Combine the results: $\int_{0}^{5} \left(3x^2 + 3\right) dx = 125 + 15 = 140$

Final Answer:

$\boxed{140}$

Let me know if you'd like more details or have any questions!

1. What are the steps to calculate definite integrals?
2. How do we split and handle polynomial integrals?
3. Why do constant terms simplify so easily in integration?
4. How does the Fundamental Theorem of Calculus apply here?
5. Could the limits be negative, and how would that change the calculation?

Tip: Always double-check each term's evaluation when applying definite limits to avoid sign errors!