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Let's go through the given questions step-by-step.

### 9.1 Determine the integral:
1. **9.1.1**: \(\int 12 \, dx\)
   - This is a straightforward integral of a constant.
   \[
   \int 12 \, dx = 12x + C
   \]

2. **9.1.2**: \(\int (x + 2)(x - 7) \, dx\)
   - First, expand the integrand:
   \[
   (x + 2)(x - 7) = x^2 - 7x + 2x - 14 = x^2 - 5x - 14
   \]
   - Now, integrate each term separately:
   \[
   \int (x^2 - 5x - 14) \, dx = \int x^2 \, dx - \int 5x \, dx - \int 14 \, dx
   \]
   \[
   = \frac{x^3}{3} - \frac{5x^2}{2} - 14x + C
   \]

### 9.2 The diagram shows the function \( f(x) = 5^x \).

1. **9.2.1** Determine \(\int 5^x \, dx\).

   The integral of \(5^x\) is given by:
   \[
   \int 5^x \, dx = \frac{5^x}{\ln(5)} + C
   \]

2. **9.2.2** Show whether \( A_1 \times 160 \) is less than the area of \( A_2 \).

   - Calculate the area \(A_1\) (from \(-1\) to \(0\)) and \(A_2\) (from \(1\) to \(3\)) by integrating \( f(x) = 5^x \) over these intervals.
   - Then compare \( A_1 \times 160 \) with \( A_2 \).

I will now calculate these areas.### Calculations:

1. **Area \( A_1 \):**  
   \[
   A_1 = \int_{-1}^{0} 5^x \, dx = \frac{4}{5 \ln(5)}
   \]

2. **Area \( A_2 \):**  
   \[
   A_2 = \int_{1}^{3} 5^x \, dx = \frac{120}{\ln(5)}
   \]

3. **Comparison:**

   - \( A_1 \times 160 = \frac{4 \times 160}{5 \ln(5)} = \frac{640}{5 \ln(5)} \)
   - Since \(\frac{640}{5 \ln(5)}\) is **not less than** \(\frac{120}{\ln(5)}\), the statement that \(A_1 \times 160\) is less than \(A_2\) is **false**.

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

Here are 5 related questions you might find interesting:
1. How do you find the area under a curve for any general function?
2. What is the significance of the natural logarithm in integration involving exponential functions?
3. Can we use numerical methods to approximate the areas \(A_1\) and \(A_2\)?
4. How would changing the base of the exponential function affect the integral result?
5. What happens to the integral if the function \(f(x)\) were a decaying exponential instead?

**Tip:** For exponential functions of the form \(a^x\), remember that their integral is \(\frac{a^x}{\ln(a)}\), which is crucial for simplifying complex exponential integrals.

9.1 Determine the integral: (9.1.1) ∫12 dx, (9.1.2) ∫(x + 2)(x - 7) dx. 9.2 Diagram shows function f(x) = 5^x, (9.2.1) Determine ∫5^x dx, (9.2.2) Show if A_1 × 160 is less than A_2.

Learn to calculate integrals of polynomial and exponential functions such as f(x) = 5^x, including detailed steps. This problem covers integration techniques and exponential growth properties. Suitable for Grades 11-12 students.

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