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The integral shown is:

\[
\int \frac{1 + \sec^3 x}{\cot x} \, dx
\]

To solve this integral, we can first simplify it by expressing \(\cot x\) in terms of \(\sin x\) and \(\cos x\), and then use substitution techniques if necessary.

### Step 1: Express \(\cot x\) in terms of sine and cosine.
\[
\cot x = \frac{\cos x}{\sin x}
\]

Now the integral becomes:
\[
\int \frac{(1 + \sec^3 x)}{\frac{\cos x}{\sin x}} \, dx = \int \frac{(1 + \sec^3 x) \sin x}{\cos x} \, dx
\]

### Step 2: Express \(\sec^3 x\) in terms of \(\cos x\).
Recall that \(\sec x = \frac{1}{\cos x}\), so:
\[
\sec^3 x = \frac{1}{\cos^3 x}
\]

Substitute this into the integral:
\[
\int \frac{(1 + \frac{1}{\cos^3 x}) \sin x}{\cos x} \, dx = \int \left( \frac{\sin x}{\cos x} + \frac{\sin x}{\cos^4 x} \right) dx
\]

### Step 3: Split the integral.
Now we have two simpler integrals to deal with:
\[
\int \frac{\sin x}{\cos x} \, dx + \int \frac{\sin x}{\cos^4 x} \, dx
\]

#### First integral:
\[
\int \frac{\sin x}{\cos x} \, dx = \int \tan x \, dx = \ln |\sec x| + C_1
\]

#### Second integral:
To handle the second integral, we can use the substitution \( u = \cos x \), so \( du = -\sin x \, dx \), transforming the integral into:
\[
-\int \frac{1}{u^4} \, du = \frac{1}{3u^3} + C_2 = \frac{1}{3\cos^3 x} + C_2
\]

### Step 4: Combine the results.
Now combine the results from both integrals:
\[
\ln |\sec x| - \frac{1}{3\cos^3 x} + C
\]

Thus, the solution to the integral is:

\[
\ln |\sec x| - \frac{1}{3\cos^3 x} + C
\]

Would you like any further clarification or details on any part of this solution? 

Here are 5 related questions to explore further:
1. How do you approach integrals involving trigonometric identities?
2. What are common trigonometric substitutions in integrals?
3. How can we integrate \(\sec^n x\) for different powers of \(n\)?
4. What is the role of logarithmic functions in trigonometric integrals?
5. Can you derive the integral of \(\tan x\) without using substitution?

**Tip:** When solving integrals involving trigonometric functions, it's helpful to recall basic identities and be comfortable using substitutions for simplification.

This page provides a detailed step-by-step solution to the integral ∫ (1 + sec^3(x)) / cot(x) dx. Key concepts include trigonometric identities, substitution methods, and integration of secant and tangent functions. Suitable for undergraduate calculus students.

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