To integrate the function \( f(x) = x^2 \) step by step, follow these instructions:

### Step 1: Identify the integral
We want to find the indefinite integral of \( f(x) = x^2 \), which means we are looking for the antiderivative of \( x^2 \). The general form of the integral is:

\[
\int x^2 \, dx
\]

### Step 2: Apply the power rule for integration
The power rule for integration states:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]

Where \( n \) is any real number, and \( C \) is the constant of integration.

For \( f(x) = x^2 \), \( n = 2 \).

### Step 3: Increase the power by 1
Using the power rule, we increase the exponent by 1:

\[
x^2 \to x^{2+1} = x^3
\]

### Step 4: Divide by the new exponent
Now divide by the new exponent (3):

\[
\int x^2 \, dx = \frac{x^3}{3}
\]

### Step 5: Add the constant of integration
The result of an indefinite integral includes a constant of integration, \( C \), because the derivative of any constant is zero. So, the final answer is:

\[
\int x^2 \, dx = \frac{x^3}{3} + C
\]

### Final Answer:
\[
\int x^2 \, dx = \frac{x^3}{3} + C
\]

Would you like more details or have any questions?

Here are 5 related questions to further explore integration:
1. How do you integrate a function like \( f(x) = x^n \) for any \( n \)?
2. How do you solve definite integrals, for example, \( \int_0^1 x^2 \, dx \)?
3. What are some applications of integrating polynomial functions?
4. How do you integrate trigonometric functions like \( \sin(x) \) or \( \cos(x) \)?
5. What is the relationship between differentiation and integration?

**Tip**: Always remember to add the constant \( C \) when calculating indefinite integrals.

Follow this detailed guide to integrating f(x) = x^2 step by step using the power rule. Learn how to apply key calculus concepts and find the antiderivative of polynomials, suitable for students in Grades 10-12.

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