Evaluate the following Integrals.
`int (dx)/(cos x (sin x + 2 cosx))`

Answer

To evaluate the integral

\[ I = \int \frac{dx}{\cos x (\sin x + 2 \cos x)}, \]

we will follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form:

\[ I = \int \frac{dx}{\cos x (\sin x + 2 \cos x)}. \]

### Step 2: Factor Out Cosine
Next, we factor out \(\cos x\) from the denominator:

\[ I = \int \frac{dx}{\cos x \sin x + 2 \cos^2 x}. \]

### Step 3: Simplify the Denominator
We can express \(\sin x\) in terms of \(\tan x\):

\[ \sin x = \tan x \cos x \implies \sin x + 2 \cos x = \tan x \cos x + 2 \cos x = \cos x (\tan x + 2). \]

Thus, we can rewrite the integral as:

\[ I = \int \frac{dx}{\cos^2 x (\tan x + 2)}. \]

### Step 4: Substitute for \(\tan x\)
Using the identity \(\sec^2 x = 1 + \tan^2 x\), we can express the integral in terms of \(\tan x\):

Let \(t = \tan x + 2\). Then, differentiating gives:

\[ dt = \sec^2 x \, dx \implies dx = \frac{dt}{\sec^2 x}. \]

### Step 5: Substitute into the Integral
Substituting \(dx\) and \(\tan x\) into the integral, we have:

\[ I = \int \frac{\sec^2 x \, dt}{\cos^2 x t} = \int \frac{dt}{t}. \]

### Step 6: Integrate
Now, we can integrate:

\[ I = \ln |t| + C = \ln |\tan x + 2| + C. \]

### Final Answer
Thus, the evaluated integral is:

\[ \int \frac{dx}{\cos x (\sin x + 2 \cos x)} = \ln |\tan x + 2| + C. \]

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