Integral

Step 1: Analyze Statement (A) Statement (A) is written as: We know the Leibniz Rule for differentiating an integral: So, the correct expression should be: 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: The derivative of an indefinite integral is simply . The constant of integration is irrelevant when differentiating. Therefore, the correct expression is: Statement (B) incorrectly adds the constant , which is unnecessary.
Step 3: Final Conclusion Both statements (A) and (B) are false, hence the correct answer is:

To evaluate the integral , we use the substitution .
Then, , and the limits change as follows: when , ; when , . Also, , and . The integral becomes: Let . Consider the substitution , so , and the limits reverse: when , ; when , . Thus: Using the identity , we get: Thus: So: The second integral is , so: Solving for : Thus, the value of the integral is: