The problem presented asks to evaluate the integral:
$\int_{0}^{1} \left(x^2 + 2x + 1\right) dx$

Let us solve it step by step:

Step 1: Expand the integrand

The expression $x^2 + 2x + 1$ can be rewritten as: $(x + 1)^2$ However, for simplicity in integration, we can leave it as $x^2 + 2x + 1$.

Step 2: Integrate term by term

Using the rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, apply integration to each term:

$\int \left(x^2 + 2x + 1\right) dx = \int x^2 dx + \int 2x dx + \int 1 dx$

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

So, the indefinite integral becomes: $\frac{x^3}{3} + x^2 + x$

Step 3: Apply the limits of integration

Now, evaluate the definite integral from $0$ to $1$: $\int_{0}^{1} \left(x^2 + 2x + 1\right) dx = \left[\frac{x^3}{3} + x^2 + x \right]_0^1$

Substitute $x = 1$ into the expression: $\frac{1^3}{3} + 1^2 + 1 = \frac{1}{3} + 1 + 1 = \frac{1}{3} + 2 = \frac{7}{3}$

Substitute $x = 0$ into the expression: $\frac{0^3}{3} + 0^2 + 0 = 0$

Step 4: Subtract the results

$\int_{0}^{1} \left(x^2 + 2x + 1\right) dx = \frac{7}{3} - 0 = \frac{7}{3}$

Final Answer:

The value of the integral is: $\boxed{\frac{7}{3}}$

Do you want further clarification on the steps?

Here are some related questions to consider:

1. How do you apply the power rule of integration for higher powers?
2. What is the importance of definite limits in integrals?
3. Can you explain why constants like $2x$ are separated during integration?
4. How do we check our work when integrating polynomials?
5. How is this method applied to more complex polynomials or functions?

Tip:

Always simplify the integrand where possible to reduce computational errors and make integration easier.