To solve the integral:

$\int_{0}^{2} (x^2 + x + 1) \, dx$

We will compute it step-by-step:

Step 1: Integrate the function

The integral of each term is computed separately:

1. $\int x^2 \, dx = \frac{x^3}{3}$
2. $\int x \, dx = \frac{x^2}{2}$
3. $\int 1 \, dx = x$

Thus, the integral becomes:

$\int (x^2 + x + 1) \, dx = \frac{x^3}{3} + \frac{x^2}{2} + x$

Step 2: Evaluate the definite integral

Now substitute the limits $x = 2$ and $x = 0$ into the antiderivative:

At $x = 2$: $\frac{(2)^3}{3} + \frac{(2)^2}{2} + 2 = \frac{8}{3} + \frac{4}{2} + 2 = \frac{8}{3} + 2 + 2 = \frac{8}{3} + 4 = \frac{20}{3}$

At $x = 0$: $\frac{(0)^3}{3} + \frac{(0)^2}{2} + 0 = 0$

Step 3: Subtract

The result of the definite integral is: $\frac{20}{3} - 0 = \frac{20}{3}$

Final Answer:

$\int_{0}^{2} (x^2 + x + 1) \, dx = \frac{20}{3}$

Would you like further steps explained or related questions discussed?

Related Questions:

1. How do definite integrals differ from indefinite integrals?
2. What are the applications of polynomial integration in real-world problems?
3. How do we handle integrals with variable limits?
4. What happens if a constant multiplier is added to the polynomial?
5. How can integrals be solved using numerical methods?

Tip:

When integrating polynomials, ensure each term is handled separately, and double-check your limits during evaluation to avoid errors.