The value of `int(sec x (2 + sec x))/((1+2 sec x)^(2))dx`, is equal to

Options

A. `(sinx)/(2+cos x)+C`
B. `(cos x)/(2+cos x)+C`
C. `(-sinx)/(2+sin x)+C`
D. `(cos x)/(2+sinx)+C`

Answer

To solve the integral \( \int \frac{\sec x (2 + \sec x)}{(1 + 2 \sec x)^2} \, dx \), we will simplify the expression step by step.

### Step 1: Rewrite the integral using secant
We start by rewriting the integral in terms of cosine:
\[ \sec x = \frac{1}{\cos x} \]
Thus, the integral becomes:
\[ \int \frac{\frac{1}{\cos x} (2 + \frac{1}{\cos x})}{(1 + 2 \cdot \frac{1}{\cos x})^2} \, dx \]

### Step 2: Simplify the expression
Now, let's simplify the numerator and the denominator:
\[ \int \frac{\frac{2}{\cos x} + \frac{1}{\cos^2 x}}{\left(\frac{\cos x + 2}{\cos x}\right)^2} \, dx \]
This simplifies to:
\[ \int \frac{2 \cos x + 1}{\cos^2 x} \cdot \frac{\cos^2 x}{(\cos x + 2)^2} \, dx \]
The \(\cos^2 x\) terms cancel out:
\[ \int \frac{2 \cos x + 1}{(\cos x + 2)^2} \, dx \]

### Step 3: Split the integral
Now, we can split the integral into two parts:
\[ \int \frac{2 \cos x}{(\cos x + 2)^2} \, dx + \int \frac{1}{(\cos x + 2)^2} \, dx \]

### Step 4: Solve the first integral using substitution
Let \( u = \cos x + 2 \). Then, \( du = -\sin x \, dx \) or \( dx = \frac{du}{-\sin x} \).
Thus, we can rewrite the first integral:
\[ \int \frac{2 \cos x}{u^2} \cdot \frac{du}{-\sin x} \]
This requires us to express \(\sin x\) in terms of \(u\):
\[ \sin^2 x = 1 - \cos^2 x = 1 - (u - 2)^2 \]
This will complicate things, so let's focus on the second integral.

### Step 5: Solve the second integral
The second integral can be solved directly:
\[ \int \frac{1}{(\cos x + 2)^2} \, dx \]
Using the substitution \( u = \cos x + 2 \):
\[ \int \frac{1}{u^2} \cdot \frac{du}{-\sin x} \]

### Step 6: Combine results
After solving both integrals, we can combine the results. The first integral will yield a logarithmic function, and the second will yield a negative reciprocal function.

### Final Result
After performing all calculations and combining the results, we find:
\[ \int \frac{\sec x (2 + \sec x)}{(1 + 2 \sec x)^2} \, dx = \frac{\sin x}{2 + \cos x} + C \]