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### Problem 10:
Find the indefinite integral \(\int (9x^2 + 4x - 3) \, dx\).

To solve this:
1. \(\int 9x^2 \, dx = 9 \cdot \frac{x^3}{3} = 3x^3\)
2. \(\int 4x \, dx = 4 \cdot \frac{x^2}{2} = 2x^2\)
3. \(\int -3 \, dx = -3x\)

Putting these together:
\[
\int (9x^2 + 4x - 3) \, dx = 3x^3 + 2x^2 - 3x + C
\]

Answer: The correct option is **D**: \(3x^3 + 2x^2 - 3x + C\).

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### Problem 11:
Find the definite integral \(\int_1^2 (6x^2 + 2x - 1) \, dx\).

Step-by-step:
1. Integrate \(6x^2\): \(\int 6x^2 \, dx = 6 \cdot \frac{x^3}{3} = 2x^3\)
2. Integrate \(2x\): \(\int 2x \, dx = x^2\)
3. Integrate \(-1\): \(\int -1 \, dx = -x\)

The indefinite integral becomes:
\[
2x^3 + x^2 - x
\]

Now, evaluate it from 1 to 2:
\[
\left[ 2(2)^3 + (2)^2 - 2 \right] - \left[ 2(1)^3 + (1)^2 - 1 \right]
\]
\[
= \left[ 16 + 4 - 2 \right] - \left[ 2 + 1 - 1 \right]
\]
\[
= (18) - (2) = 16
\]

Answer: The correct option is **E**: 16.

---

### Problem 12:
Find the definite integral \(\int_{\frac{\pi}{2}}^{\pi} \sin x \, dx\).

Step-by-step:
1. The integral of \(\sin x\) is \(-\cos x\).
2. Now, evaluate it from \(\frac{\pi}{2}\) to \(\pi\):
\[
-\cos(\pi) - (-\cos(\frac{\pi}{2})) = -(-1) - (0) = 1
\]

Answer: The correct option is **C**: 1.

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Would you like further details or explanations?

Here are some related questions to expand on:
1. What is the relationship between definite and indefinite integrals?
2. How do we apply the Fundamental Theorem of Calculus in definite integrals?
3. Can you explain how to handle integration by substitution?
4. What are the common trigonometric integrals and their antiderivatives?
5. How does one approach solving integrals of polynomial functions?

**Tip:** Always check the limits of integration carefully when solving definite integrals!

Find the indefinite integral of (9x^2 + 4x - 3) and two definite integrals: one for (6x^2 + 2x - 1) from 1 to 2 and the other for sin(x) from π/2 to π.

Learn how to solve integrals of polynomial and trigonometric functions. This includes the indefinite integral of (9x^2 + 4x - 3), and definite integrals from 1 to 2 of (6x^2 + 2x - 1) and from π/2 to π of sin(x). Suitable for Grade 11-12 students.

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