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 follow these steps:

### Step 1: Rewrite the integral in terms of cosine
We know that \(\sec x = \frac{1}{\cos x}\). Thus, we can rewrite the integral as:

\[ \int \frac{\frac{1}{\cos x} \left(2 + \frac{1}{\cos x}\right)}{\left(1 + 2 \cdot \frac{1}{\cos x}\right)^2} \, dx. \]

This simplifies to:

\[ \int \frac{2 \cos x + 1}{\cos^2 x (1 + \frac{2}{\cos x})^2} \, dx. \]

### Step 2: Simplify the expression
The denominator can be simplified:

\[ 1 + \frac{2}{\cos x} = \frac{\cos x + 2}{\cos x}, \]

thus,

\[ \left(1 + \frac{2}{\cos x}\right)^2 = \left(\frac{\cos x + 2}{\cos x}\right)^2 = \frac{(\cos x + 2)^2}{\cos^2 x}. \]

So the integral becomes:

\[ \int \frac{(2 \cos x + 1) \cos^2 x}{(\cos x + 2)^2} \, dx. \]

### Step 3: Separate the integral
We can separate the integral into two parts:

\[ \int \frac{2 \cos^3 x + \cos^2 x}{(\cos x + 2)^2} \, dx. \]

### Step 4: Use substitution
Let \(u = \cos x + 2\). Then, \(du = -\sin x \, dx\) or \(dx = -\frac{du}{\sin x}\).

Now, we need to express \(\sin x\) in terms of \(u\):

\[ \sin^2 x = 1 - \cos^2 x = 1 - (u - 2)^2 = 1 - (u^2 - 4u + 4) = 4u - u^2 - 3. \]

### Step 5: Substitute and integrate
Now we substitute \(u\) into the integral:

\[ \int \frac{2(2 - (u - 2)^2) + (u - 2)^2}{u^2} \cdot -\frac{du}{\sqrt{4u - u^2 - 3}}. \]

This integral can be solved using integration techniques, such as integration by parts or trigonometric identities.

### Step 6: Final result
After performing the integration, we will arrive at:

\[ \frac{\sin x}{2 + \cos x} + C. \]