The value of `int_(-2)^(2) (ax^(3) + bx+ c) dx` depends on which following ?

Options

A. Values of x only
B. Values of each of a,b and c
C. Value of c only
D. Value of b only

Explanation

To solve the integral \( \int_{-2}^{2} (ax^3 + bx + c) \, dx \) and determine its dependence, we will break down the integral into parts and evaluate each component.

### Step 1: Break down the integral
We can separate the integral into three parts:
\[ \int_{-2}^{2} (ax^3 + bx + c) \, dx = \int_{-2}^{2} ax^3 \, dx + \int_{-2}^{2} bx \, dx + \int_{-2}^{2} c \, dx \]

### Step 2: Evaluate each integral
1. **Evaluate \( \int_{-2}^{2} ax^3 \, dx \)**:
- The function \( ax^3 \) is an odd function (since \( x^3 \) is odd).
- The integral of an odd function over a symmetric interval around zero is zero:
\[ \int_{-2}^{2} ax^3 \, dx = 0 \]

2. **Evaluate \( \int_{-2}^{2} bx \, dx \)**:
- The function \( bx \) is also an odd function.
- Similarly, the integral of an odd function over a symmetric interval is zero:
\[ \int_{-2}^{2} bx \, dx = 0 \]

3. **Evaluate \( \int_{-2}^{2} c \, dx \)**:
- The function \( c \) is a constant function.
- The integral of a constant \( c \) over the interval from \(-2\) to \(2\) is:
\[ \int_{-2}^{2} c \, dx = c \cdot (2 - (-2)) = c \cdot 4 = 4c \]

### Step 3: Combine the results
Now, combining all the results, we have:
\[ \int_{-2}^{2} (ax^3 + bx + c) \, dx = 0 + 0 + 4c = 4c \]

### Conclusion
The value of the integral \( \int_{-2}^{2} (ax^3 + bx + c) \, dx \) depends solely on the constant \( c \).

### Final Answer
Thus, the value of the integral depends on \( c \).

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