Assertion (A) : `int x^(2) dx = (x^(3))/(3) + c`
Reason (R) : `int e^xdx = xe^x + c`

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 question, we need to analyze the assertion and the reason provided.

**Assertion (A):**
\[ \int x^2 \, dx = \frac{x^3}{3} + C \]

**Reason (R):**
\[ \int e^x \, dx = x e^x + C \]

### Step-by-Step Solution:

1. **Evaluate the Assertion (A):**
- We know the formula for integrating a power function:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1 \]
- For \( n = 2 \):
\[ \int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C \]
- Therefore, Assertion (A) is **true**.

2. **Evaluate the Reason (R):**
- The integral of \( e^x \) is given by:
\[ \int e^x \, dx = e^x + C \]
- The statement in Reason (R) claims that:
\[ \int e^x \, dx = x e^x + C \]
- This is incorrect because the correct integral of \( e^x \) does not include the \( x e^x \) term.
- Therefore, Reason (R) is **false**.

3. **Conclusion:**
- Since Assertion (A) is true and Reason (R) is false, the correct conclusion is that Assertion (A) is correct, but Reason (R) is incorrect.

### Final Answer:
- Assertion (A) is true, and Reason (R) is false.