Evaluate `int "cosec"^(4)x dx`

Explanation

To evaluate the integral \( \int \csc^4 x \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We can express \( \csc^4 x \) in terms of \( \csc^2 x \):
\[ \csc^4 x = \csc^2 x \cdot \csc^2 x \]
Thus, we rewrite the integral as:
\[ \int \csc^4 x \, dx = \int \csc^2 x \cdot \csc^2 x \, dx \]

### Step 2: Use the Identity for \( \csc^2 x \)
We know that:
\[ \csc^2 x = 1 + \cot^2 x \]
So, we can substitute this into our integral:
\[ \int \csc^4 x \, dx = \int (1 + \cot^2 x) \csc^2 x \, dx \]

### Step 3: Distribute the Integral
Now we can separate the integral:
\[ \int \csc^4 x \, dx = \int \csc^2 x \, dx + \int \cot^2 x \csc^2 x \, dx \]

### Step 4: Evaluate the First Integral
The integral of \( \csc^2 x \) is a standard integral:
\[ \int \csc^2 x \, dx = -\cot x \]

### Step 5: Evaluate the Second Integral
For the second integral \( \int \cot^2 x \csc^2 x \, dx \), we can use the substitution:
Let \( t = \cot x \), then \( dt = -\csc^2 x \, dx \) or \( dx = -\frac{dt}{\csc^2 x} \).

Thus, we rewrite the integral:
\[ \int \cot^2 x \csc^2 x \, dx = -\int t^2 \, dt \]

### Step 6: Integrate
Now we integrate:
\[ -\int t^2 \, dt = -\frac{t^3}{3} + C = -\frac{\cot^3 x}{3} + C \]

### Step 7: Combine the Results
Now we combine the results of both integrals:
\[ \int \csc^4 x \, dx = -\cot x - \frac{\cot^3 x}{3} + C \]

### Final Answer
Thus, the final answer is:
\[ \int \csc^4 x \, dx = -\cot x - \frac{\cot^3 x}{3} + C \]

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