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

Options

A. B
B. C
C. A
D. none of these

Explanation

To solve the integral \( \int_{-2}^{2} (ax^3 + bx + c) \, dx \), we will follow these steps:

### Step 1: Set Up the Integral
We start with the integral:
\[ \int_{-2}^{2} (ax^3 + bx + c) \, dx \]

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

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

2. **Evaluate \( \int_{-2}^{2} bx \, dx \)**:
Similarly, \( bx \) is also an odd function, so:
\[ \int_{-2}^{2} bx \, dx = 0 \]

3. **Evaluate \( \int_{-2}^{2} c \, dx \)**:
The term \( c \) is a constant, so we can factor it out:
\[ \int_{-2}^{2} c \, dx = c \int_{-2}^{2} 1 \, dx = c \cdot [x]_{-2}^{2} = c \cdot (2 - (-2)) = c \cdot 4 = 4c \]

### Step 4: Combine the Results
Now, we combine the results from each part:
\[ \int_{-2}^{2} (ax^3 + bx + c) \, dx = 0 + 0 + 4c = 4c \]

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

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

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