f(x) = `int (dx)/(sin^(4)x)` is a

Answer

To find the function \( f(x) = \int \frac{dx}{\sin^4 x} \), we will proceed with the integration step by step.

### Step 1: Rewrite the Integral
We start with the integral:
\[ f(x) = \int \frac{dx}{\sin^4 x} \]
We can rewrite this using the identity \( \sin^2 x = 1 - \cos^2 x \):
\[ f(x) = \int \csc^4 x \, dx \]

### Step 2: Use the Identity for Cosecant
Recall that \( \csc^2 x = 1 + \cot^2 x \). Thus, we can express \( \csc^4 x \) as:
\[ \csc^4 x = (\csc^2 x)^2 = (1 + \cot^2 x)^2 \]
Expanding this gives:
\[ \csc^4 x = 1 + 2\cot^2 x + \cot^4 x \]

### Step 3: Substitute into the Integral
Now we substitute this back into the integral:
\[ f(x) = \int (1 + 2\cot^2 x + \cot^4 x) \, dx \]
This can be split into three separate integrals:
\[ f(x) = \int 1 \, dx + 2 \int \cot^2 x \, dx + \int \cot^4 x \, dx \]

### Step 4: Integrate Each Term
1. The first integral:
\[ \int 1 \, dx = x \]

2. The second integral, using the identity \( \cot^2 x = \csc^2 x - 1 \):
\[ \int \cot^2 x \, dx = \int (\csc^2 x - 1) \, dx = -\cot x - x \]
Therefore,
\[ 2 \int \cot^2 x \, dx = -2\cot x - 2x \]

3. The third integral \( \int \cot^4 x \, dx \) can be computed using the reduction formula or integration by parts, but for simplicity, we can use the known result:
\[ \int \cot^4 x \, dx = \frac{1}{3} \cot^3 x - \cot x + C \]

### Step 5: Combine All Parts
Putting all the parts together, we have:
\[ f(x) = x - 2\cot x - 2x + \frac{1}{3} \cot^3 x - \cot x + C \]
Combining like terms:
\[ f(x) = -x - 3\cot x + \frac{1}{3} \cot^3 x + C \]

### Conclusion
Thus, the function \( f(x) \) is a polynomial in terms of \( \cot x \) of degree 3.