Definite Integral

Step 1: Compute values of integrals

For :

For :

For :
But more directly:

Step 2: Check for Arithmetic Progression (A.P.)

Step 3: Check for Geometric Progression (G.P.)

Step 4: Check for Harmonic Progression (H.P.)

Conclusion:
The sequence is in a Geometric Progression (G.P.).

Given Information

The correct answer is .

The mean value is given by:

Applying the Formula

In this specific case, we are calculating the mean value of over the interval . Therefore, and .

Calculating the Integral

Now, we evaluate the integral:

Evaluating the Limits

Substitute the limits of integration:

Finding the Mean Value

Substitute the value of the integral back into the formula for :

Conclusion

Therefore, the mean value of over the interval is .

Integral Symmetry and Evaluation

Integral Equality: The equality holds when the function satisfies .

Zero Integral Equivalence: The integral if the function satisfies .

Given:

To test symmetry, evaluate :

Conclusion: Since , the function is odd about , and the integral over is therefore:

Final Answer: Option (D): 0