To integrate the expression \( 4 \csc^4(x) \cot^6(x) \), we can approach it by expressing the trigonometric functions in terms of basic identities and performing a substitution if necessary. Let's go step by step:

### Step 1: Rewrite using trigonometric identities
We know the following trigonometric identities:

- \( \csc(x) = \frac{1}{\sin(x)} \)
- \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

Thus, the given expression becomes:

\[
4 \csc^4(x) \cot^6(x) = 4 \left( \frac{1}{\sin^4(x)} \right) \left( \frac{\cos^6(x)}{\sin^6(x)} \right) = 4 \frac{\cos^6(x)}{\sin^{10}(x)}
\]

### Step 2: Simplify and prepare for substitution
We now have the expression in a simplified form:

\[
4 \frac{\cos^6(x)}{\sin^{10}(x)}
\]

Since the powers of \( \sin(x) \) and \( \cos(x) \) are quite large, we can use the substitution \( u = \sin(x) \), which means \( du = \cos(x) dx \). Let's rewrite the integral in terms of \( u \).

### Step 3: Perform substitution
Given \( u = \sin(x) \), we have \( du = \cos(x) dx \). The expression \( \cos^6(x) \) can be split into:

\[
\cos^6(x) = (\cos^2(x))^3 = (1 - \sin^2(x))^3 = (1 - u^2)^3
\]

So, the integral becomes:

\[
\int 4 \frac{\cos^6(x)}{\sin^{10}(x)} dx = 4 \int \frac{(1 - u^2)^3}{u^{10}} du
\]

### Step 4: Expand and integrate
Now, expand \( (1 - u^2)^3 \):

\[
(1 - u^2)^3 = 1 - 3u^2 + 3u^4 - u^6
\]

So the integral becomes:

\[
4 \int \frac{1 - 3u^2 + 3u^4 - u^6}{u^{10}} du = 4 \int \left( u^{-10} - 3u^{-8} + 3u^{-6} - u^{-4} \right) du
\]

### Step 5: Integrate term by term
Now, integrate each term:

\[
\int u^{-10} du = \frac{u^{-9}}{-9} = -\frac{1}{9u^9}
\]
\[
\int u^{-8} du = \frac{u^{-7}}{-7} = -\frac{1}{7u^7}
\]
\[
\int u^{-6} du = \frac{u^{-5}}{-5} = -\frac{1}{5u^5}
\]
\[
\int u^{-4} du = \frac{u^{-3}}{-3} = -\frac{1}{3u^3}
\]

Thus, the integral becomes:

\[
4 \left( -\frac{1}{9u^9} + \frac{3}{7u^7} - \frac{3}{5u^5} + \frac{1}{3u^3} \right) + C
\]

### Step 6: Substitute \( u = \sin(x) \) back
Now, substitute \( u = \sin(x) \) back into the expression:

\[
4 \left( -\frac{1}{9\sin^9(x)} + \frac{3}{7\sin^7(x)} - \frac{3}{5\sin^5(x)} + \frac{1}{3\sin^3(x)} \right) + C
\]

This is the final integrated result.

---

Would you like further explanation on any of these steps?

Here are five related questions you might be interested in:
1. How do you handle more complex trigonometric integrals?
2. What is the method of substitution in calculus?
3. How can we integrate functions involving powers of cotangent and cosecant?
4. What are some typical applications of trigonometric integrals in physics?
5. How do we approach integrals involving mixed trigonometric identities?

**Tip:** When facing integrals with high powers of trigonometric functions, always try simplifying using identities or substitution.

Learn how to integrate the complex trigonometric expression 4csc^4(x) cot^6(x) using substitution and trigonometric identities. This step-by-step guide simplifies the integral by using basic trig identities and substitution. Suitable for students at the undergraduate calculus level or advanced high school calculus learners.

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