Integral Calculus

Step 1: Identifying the symmetry.
The given integral involves an odd function ( ) integrated over a symmetric interval from to . The integral of an odd function over a symmetric interval is always 0.

Step 2: Analyzing the options.
(A) 0: Correct — The integral of an odd function over a symmetric interval is zero.
(B) : Incorrect — This is not the correct value of the integral.
(C) 1: Incorrect — The result is not 1.
(D) 2: Incorrect — This is not the correct value of the integral.

Step 3: Conclusion.
The correct answer is (A) 0 because the integrand is an odd function, and its integral over a symmetric interval is zero.

Step 1: Analyze the function.
The given function is . To find its maximum value, we need to take the first derivative, , and set it equal to zero.
Step 2: Differentiate the function.
Using the product rule of differentiation, we get: Simplify: Step 3: Set the derivative equal to zero to find the critical points.
Since for any , we have: Factor: This gives us two critical points: Step 4: Verify which critical point gives the maximum value.
To determine whether or gives the maximum, we can check the second derivative or simply evaluate at both points.
- At , .
- At , .
Thus, the maximum value of is .
Final Answer: