https://file.aiquickdraw.com/mathbot/file/172633130942320641a57.jpg

The given image shows the following integral:

\[
\int_1^2 (x^2 + x) \, dx
\]

### Step-by-step solution:

1. **Separate the integral:**

   The integral can be split into two parts:

   \[
   \int_1^2 (x^2 + x) \, dx = \int_1^2 x^2 \, dx + \int_1^2 x \, dx
   \]

2. **Solve each integral:**

   - For \( \int x^2 \, dx \), the antiderivative is \( \frac{x^3}{3} \).
   - For \( \int x \, dx \), the antiderivative is \( \frac{x^2}{2} \).

3. **Evaluate both integrals:**

   Now, apply the limits of integration (from 1 to 2):

   \[
   \int_1^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_1^2 = \frac{2^3}{3} - \frac{1^3}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}
   \]

   \[
   \int_1^2 x \, dx = \left[ \frac{x^2}{2} \right]_1^2 = \frac{2^2}{2} - \frac{1^2}{2} = \frac{4}{2} - \frac{1}{2} = 2 - \frac{1}{2} = \frac{3}{2}
   \]

4. **Add the results:**

   \[
   \int_1^2 (x^2 + x) \, dx = \frac{7}{3} + \frac{3}{2}
   \]

   To add these fractions, find a common denominator:

   \[
   \frac{7}{3} + \frac{3}{2} = \frac{14}{6} + \frac{9}{6} = \frac{23}{6}
   \]

### Final answer:

\[
\int_1^2 (x^2 + x) \, dx = \frac{23}{6}
\]

Would you like more details on any step or have any questions?

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Here are 5 related questions to explore further:

1. What is the geometric interpretation of a definite integral?
2. How would the result change if the limits of integration were different?
3. Can you generalize the formula for integrating polynomials?
4. What is the Fundamental Theorem of Calculus, and how does it apply here?
5. How can you verify the result using numerical methods?

**Tip:** When solving definite integrals, always double-check by evaluating the antiderivatives at both limits to avoid small calculation errors.

Find the definite integral of (x^2 + x) dx from 1 to 2.

Learn how to solve the definite integral of the polynomial expression (x^2 + x) from 1 to 2. This step-by-step solution covers key calculus concepts, including polynomial integration and the Fundamental Theorem of Calculus. Suitable for high school students in Grades 10-12.

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