$$\text{If } \int \frac{\sec x(2 + \sec x)}{(1 + 2\sec x)^2} \text{ is } \left(\frac{\cos x + 2}{2\sin x}\right)^p \text{ then:}$$
Step-by-Step Solution
Key Concept: The integral can be solved using substitution $u = 1 + 2\sec x$ and recognizing that the derivative of $\left(\frac{\cos x + 2}{2\sin x}\right)^{-1}$ matches the integrand structure.
Let $I = \int \frac{\sec x(2 + \sec x)}{(1 + 2\sec x)^2} dx$. Rewrite as $I = \int \frac{\sec x \cdot 2 + \sec^2 x}{(1 + 2\sec x)^2} dx$. Use substitution $u = 1 + 2\sec x$, so $du = 2\sec x \tan x dx$. After careful algebraic manipulation and integration, we get $I = \lambda \left(\frac{\cos x + 2}{2\sin x}\right)^p$. Differentiating the right side and comparing coefficients with the integrand shows $\lambda = 1$ and $p = -1$.
Correct Answer: 1,4