The given integral is:
We need to solve this to find its value in terms of . To simplify the process and identify the structure, let's consider the standard form of the integral involving trigonometric functions. The denominator resembles the structure of a transformed trigonometric identity:
The expression can be rewritten using the identity for the cosine of a double angle, , as part of completing the square:
Recognizing the structure of an integrable form, we use identities and symmetry to transform it. Let us try the substitutions and simplification:
This integral can be evaluated using a known result involving trigonometric integrals:
Given the bounds from 0 to and , this result simplifies to:
This follows from reducing the original expression to a function over a standard trigonometric range and simplifying using known integral solutions.
Thus, the value of the integral is , which corresponds to option B:
Correct Answer: