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

Options

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

Answer

To solve the problem of evaluating the definite integral \( \int_{-2}^{2} (ax^3 + bx + c) \, dx \) and determining which parameters (a, b, c) it depends on, we can follow these steps:

### Step 1: Separate 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. **For \( \int_{-2}^{2} ax^3 \, dx \)**:
- 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. **For \( \int_{-2}^{2} bx \, dx \)**:
- The function \( bx \) is also an odd function.
- Similarly, the integral of \( bx \) over a symmetric interval is zero:
\[ \int_{-2}^{2} bx \, dx = 0 \]

3. **For \( \int_{-2}^{2} c \, dx \)**:
- The function \( c \) is a constant.
- The integral of a constant 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, we can combine the results of the three integrals:
\[ \int_{-2}^{2} (ax^3 + bx + c) \, dx = 0 + 0 + 4c = 4c \]

### Conclusion
The value of the integral \( \int_{-2}^{2} (ax^3 + bx + c) \, dx \) depends only on the constant \( c \). Therefore, the correct answer is that the value of the integral depends on **c only**.