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To solve the integral \( \int 4 \csc x \cot x \, dx \), we can proceed as follows:

### Step 1: Recognize the integral form
The integral \( \int \csc x \cot x \, dx \) is a standard integral. We know that:
\[
\int \csc x \cot x \, dx = -\csc x + C
\]
where \( C \) is the constant of integration.

### Step 2: Apply the constant multiple
Given that the integral is \( \int 4 \csc x \cot x \, dx \), we multiply the result by 4:
\[
\int 4 \csc x \cot x \, dx = 4 \left( -\csc x \right) + C = -4 \csc x + C
\]

### Step 3: Match with the provided options
Comparing the result \(-4 \csc x + C\) with the options provided:

- **A)** \(-4 \cot x + C\)
- **B)** \(-4 \cos x + C\)
- **C)** \(-4 \csc x + C\)
- **D)** \(-4 \sin x + C\)

The correct answer is **C)**.

Would you like more details or have any questions? Here are five related questions that could deepen your understanding:

1. What is the integral of \( \csc^2 x \, dx \)?
2. How does the integral of \( \sec x \, dx \) compare to \( \csc x \, dx \)?
3. Can you explain the steps for integrating \( \cot x \, dx \)?
4. What are the properties of the cotangent and cosecant functions that are useful in integration?
5. How would you solve the integral \( \int \csc^3 x \, dx \)?

**Tip:** For integrals involving trigonometric functions like \(\csc x\) and \(\cot x\), it is useful to remember their standard integral forms to quickly identify the solution.

Learn how to solve the integral of 4csc(x)cot(x) with a detailed step-by-step solution. This problem covers key calculus concepts, including trigonometric integrals and integration techniques. Suitable for high school students in Grades 11-12.

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