`int(1)/((2 sin x + cos x)^(2))dx`

Answer

To solve the integral \( \int \frac{1}{(2 \sin x + \cos x)^2} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
Let \( I = \int \frac{1}{(2 \sin x + \cos x)^2} \, dx \).

### Step 2: Multiply and Divide by \( \cos^2 x \)
We can multiply and divide the integrand by \( \cos^2 x \):
\[ I = \int \frac{\cos^2 x}{(2 \sin x + \cos x)^2 \cos^2 x} \, dx \]
This gives us:
\[ I = \int \frac{\cos^2 x}{(2 \tan x + 1)^2} \, dx \]

### Step 3: Change of Variable
Let \( t = \tan x \). Then, we have:
\[ \frac{dt}{dx} = \sec^2 x \quad \Rightarrow \quad dx = \frac{dt}{\sec^2 x} = \frac{dt}{1 + t^2} \]
Substituting this into the integral, we get:
\[ I = \int \frac{1}{(2t + 1)^2} \cdot \frac{dt}{1 + t^2} \]

### Step 4: Simplify the Integral
Now we can simplify the integral:
\[ I = \int \frac{dt}{(2t + 1)^2 (1 + t^2)} \]

### Step 5: Partial Fraction Decomposition
We can use partial fraction decomposition:
\[ \frac{1}{(2t + 1)^2 (1 + t^2)} = \frac{A}{2t + 1} + \frac{B}{(2t + 1)^2} + \frac{Ct + D}{1 + t^2} \]
Multiplying through by the denominator \( (2t + 1)^2 (1 + t^2) \) and solving for constants \( A, B, C, D \).

### Step 6: Integrate Each Term
After finding the constants, we will integrate each term separately. The integral of \( \frac{1}{(2t + 1)} \) and \( \frac{1}{(1 + t^2)} \) will yield logarithmic and arctangent functions respectively.

### Step 7: Back Substitute
Finally, we substitute back \( t = \tan x \) into our results to express the integral in terms of \( x \).

### Step 8: Final Answer
The final answer will be in the form:
\[ I = -\frac{1}{2(2 \tan x + 1)} + C \]
where \( C \) is the constant of integration.

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