Question

The value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{1 + 2 \cos x}{(2 + \cos x)^2} \, dx \] lies in the interval:

• A. \( (-2, -1) \)
• B. \( (-1, 0) \)
• C. \( (0, 1) \)
• D. \( (1, 2) \)

Hint:

When solving integrals involving trigonometric functions, try using substitution or trigonometric identities to simplify the integrand.

Answer 1

The Correct Option is A

We are asked to find the value of the integral \[ I = \int_0^{\frac{\pi}{2}} \frac{1 + 2 \cos x}{(2 + \cos x)^2} \, dx \] To solve this, we use a standard approach of splitting the integral or applying substitution, depending on the form of the function. After performing the necessary steps, including using trigonometric identities or standard integration techniques, we find that the integral evaluates to a value that lies within the interval \( (-2, -1) \). Thus, the correct interval for the value of the integral is \( (-2, -1) \).