Assertion (A) : `int_(-3)^(3) (x^(3) + 5)dx = 30`
Reason (R) : `f(x) = x^(3) + 5` is an odd function.

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is NOT the correct explanation of A.
C. A is true but R is false
D. A is false and R is true

Answer

To solve the problem, we need to evaluate the integral and analyze the function to determine the validity of the assertion and reason given.

### Step-by-Step Solution:

1. **Evaluate the Integral**:
We need to compute the integral:
\[ \int_{-3}^{3} (x^3 + 5) \, dx \]

2. **Separate the Integral**:
We can separate the integral into two parts:
\[ \int_{-3}^{3} x^3 \, dx + \int_{-3}^{3} 5 \, dx \]

3. **Evaluate the First Integral**:
The first integral is:
\[ \int_{-3}^{3} x^3 \, dx \]
Since \(x^3\) is an odd function, the integral over a symmetric interval around zero is zero:
\[ \int_{-3}^{3} x^3 \, dx = 0 \]

4. **Evaluate the Second Integral**:
The second integral is:
\[ \int_{-3}^{3} 5 \, dx = 5 \int_{-3}^{3} 1 \, dx = 5 \cdot (3 - (-3)) = 5 \cdot 6 = 30 \]

5. **Combine the Results**:
Now, we combine the results of both integrals:
\[ \int_{-3}^{3} (x^3 + 5) \, dx = 0 + 30 = 30 \]

6. **Conclusion about the Assertion**:
The assertion states that:
\[ \int_{-3}^{3} (x^3 + 5) \, dx = 30 \]
This is correct.

7. **Analyze the Reason**:
The reason states that \(f(x) = x^3 + 5\) is an odd function. To check if this is true, we need to evaluate \(f(-x)\):
\[ f(-x) = (-x)^3 + 5 = -x^3 + 5 \]
This does not equal \(-f(x)\) because:
\[ -f(x) = -(x^3 + 5) = -x^3 - 5 \]
Therefore, \(f(x)\) is not an odd function.

8. **Final Conclusion**:
The assertion is true, but the reason is false. Thus, the correct option would be that the assertion is correct, but the reason is incorrect.