Assertion: Integrating factor of `(dy )/(dx) + y = x^2 ` is ` e^x`
Reason: Integrating factor of ` ( dy )/(dx) + P( x) y= Q (x) ` is ` e^(int p(x) dx)`

Options

A. If both assertion and reason are CORRECT and the reason is CORRECT explanation of the assertion.
B. If both assertion and reason are CORRECT and the reason in INCORRECT explanation of the assertion.
C. If assertion is CORRECT and the reason in INCORRECT
D. If assertion in INCORRECT and the reason is CORRECT.

Answer

To solve the problem, we need to analyze the assertion and the reason provided in the question regarding the integrating factor of the given differential equation.

### Step-by-Step Solution:

1. **Identify the Differential Equation**:
The given differential equation is:
\[ \frac{dy}{dx} + y = x^2 \]
This is in the standard form of a first-order linear differential equation, which is:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
Here, we identify:
- \( P(x) = 1 \)
- \( Q(x) = x^2 \)

2. **Determine the Integrating Factor**:
The integrating factor \( \mu(x) \) for a linear differential equation is given by:
\[ \mu(x) = e^{\int P(x) \, dx} \]
Substituting \( P(x) = 1 \):
\[ \mu(x) = e^{\int 1 \, dx} = e^{x} \]

3. **Verify the Assertion**:
The assertion states that the integrating factor of the equation is \( e^x \). From our calculation, we found that:
\[ \mu(x) = e^{x} \]
Therefore, the assertion is correct.

4. **Verify the Reason**:
The reason states that the integrating factor of \( \frac{dy}{dx} + P(x)y = Q(x) \) is \( e^{\int P(x) \, dx} \). This is indeed the general formula for finding the integrating factor of a linear first-order differential equation. Since we used this formula to find the integrating factor, the reason is also correct.

5. **Conclusion**:
Both the assertion and the reason are correct, and the reason correctly explains the assertion.

### Final Answer:
Both the assertion and the reason are correct. The integrating factor of \( \frac{dy}{dx} + y = x^2 \) is \( e^x \), and the reason provided is a valid explanation for the assertion.