Integral

We are tasked with simplifying , where is the imaginary unit, defined by .
Step 1: Simplifying We know that powers of cycle every four terms: To simplify , we divide 57 by 4 and find the remainder: Thus:
Step 2: Simplifying Next, we simplify . Again, powers of cycle every 4 terms. To simplify , we divide 25 by 4 and find the remainder: Thus:
So:
Now, we can multiply the numerator and denominator by to get rid of the imaginary unit in the denominator:

Step 3: Final Calculation Now, substitute the results back into the original expression:
Thus, the value of the expression is .
The correct answer is .

Step 1: Given:
Simplifying :
Differentiating with respect to :
Further simplification of :
Observation of :

• Positive for and
• Negative for and

Monotonicity of :

• Decreasing in
• Increasing in

Step 1: {Compute }
Using and property of floor function: Step 2: {Compare with }
Step 3: {Conclusion}
Since , the function is odd.

To evaluate the integral , let's consider a substitution method. By using Feynman's technique of differentiation under the integral sign, let . Our goal is to find .

First, differentiate with respect to :

.

Now, evaluate this new integral:

Using partial fraction decomposition: . Find , , and :

Simplifying, we equate coefficients and find that , , .

Thus,

.

The first integral is: (using substitution and trigonometric substitution).

The second integral .

Therefore,

.

Integrate with respect to over to to find :

.

This simplifies to evaluate to , matching the correct answer.