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 with the integral:
\[ I = \int \frac{dx}{\cos x (\sin x + 2 \cos x)} \]
To simplify, we divide both the numerator and the denominator by \(\cos^2 x\):
\[ I = \int \frac{1}{\cos^2 x} \cdot \frac{1}{\frac{\sin x}{\cos x} + 2} \, dx \]

### Step 2: Simplify the Integral
Using the identity \(\frac{1}{\cos^2 x} = \sec^2 x\) and \(\frac{\sin x}{\cos x} = \tan x\), we can rewrite the integral:
\[ I = \int \sec^2 x \cdot \frac{1}{\tan x + 2} \, dx \]

### Step 3: Substitution
Now, we will use the substitution \(t = \tan x + 2\). Then, the derivative of \(\tan x\) is \(\sec^2 x\), which gives us:
\[ dt = \sec^2 x \, dx \quad \Rightarrow \quad dx = \frac{dt}{\sec^2 x} \]
Substituting these into the integral, we get:
\[ I = \int \frac{1}{t} \, dt \]

### Step 4: Integrate
The integral of \(\frac{1}{t}\) is:
\[ I = \log |t| + C \]
Substituting back for \(t\):
\[ I = \log |\tan x + 2| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ I = \log (\tan x + 2) + C \]

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