Question

Consider the following statements (A) and (B): \[ \text{(A):} \quad \int_a^b \frac{d}{dx} \left( f(x) \right) dx = \frac{d}{dx} \int_a^b f(x) dx \] \[ \text{(B):} \quad \frac{d}{dx} \left( \int f(x) dx \right) = f(x) + C \] Which one of the following is True?

• A. Only (A) is true
• B. Only (B) is true
• C. Both (A) and (B) are true
• D. Both (A) and (B) are false

Hint:

When using Leibniz’s rule, remember that the derivative of an integral with fixed limits is the difference between the function evaluated at the limits. The constant of integration vanishes when differentiating indefinite integrals.

Answer 1

The Correct Option is D

Step 1: Analyze Statement (A) Statement (A) is written as: \[ \int_a^b \frac{d}{dx} \left( f(x) \right) dx = \frac{d}{dx} \int_a^b f(x) dx \] We know the Leibniz Rule for differentiating an integral: \[ \frac{d}{dx} \int_a^b f(x) \, dx = f(b) - f(a) \] So, the correct expression should be: \[ \int_a^b \frac{d}{dx} \left( f(x) \right) dx = f(b) - f(a) \] Thus, the statement (A) is incorrect, as it implies the derivative inside the integral equals the derivative of the integral over the limits. 
Step 2: Analyze Statement (B) Statement (B) is written as: \[ \frac{d}{dx} \left( \int f(x) \, dx \right) = f(x) + C \] The derivative of an indefinite integral \( \int f(x) \, dx \) is simply \( f(x) \). The constant of integration \( C \) is irrelevant when differentiating. Therefore, the correct expression is: \[ \frac{d}{dx} \left( \int f(x) \, dx \right) = f(x) \] Statement (B) incorrectly adds the constant \( C \), which is unnecessary. 
Step 3: Final Conclusion Both statements (A) and (B) are false, hence the correct answer is: \[ \boxed{\text{Both (A) and (B) are false.}} \]