Evaluate the following indefinite integral. (Use C for the constant of the antiderivative.).        integral (csc(x) + cot(x)+ 2sec(x)tan(x))dx

We are tasked with evaluating the indefinite integral:

\[
\int \left( \csc(x) + \cot(x) + 2\sec(x)\tan(x) \right) dx
\]

### Step-by-Step Solution:

1. **Break the integral into parts:**
   \[
   \int \left( \csc(x) + \cot(x) + 2\sec(x)\tan(x) \right) dx = \int \csc(x) \, dx + \int \cot(x) \, dx + \int 2\sec(x)\tan(x) \, dx
   \]

2. **Evaluate each part separately:**

   - **Integral of \( \csc(x) \):**
     \[
     \int \csc(x) \, dx = -\ln|\csc(x) + \cot(x)| + C_1
     \]

   - **Integral of \( \cot(x) \):**
     \[
     \int \cot(x) \, dx = \ln|\sin(x)| + C_2
     \]

   - **Integral of \( 2\sec(x)\tan(x) \):**
     Since \( \frac{d}{dx}[\sec(x)] = \sec(x)\tan(x) \), we have:
     \[
     \int 2\sec(x)\tan(x) \, dx = 2\sec(x) + C_3
     \]

3. **Combine the results:**
   \[
   \int \left( \csc(x) + \cot(x) + 2\sec(x)\tan(x) \right) dx = -\ln|\csc(x) + \cot(x)| + \ln|\sin(x)| + 2\sec(x) + C
   \]
   where \( C \) is the constant of integration, combining \( C_1 + C_2 + C_3 \).

Thus, the final answer is:

\[
-\ln|\csc(x) + \cot(x)| + \ln|\sin(x)| + 2\sec(x) + C
\]

Would you like further details on any of these steps?

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Here are 5 follow-up questions:

1. How do you simplify logarithmic terms in trigonometric integrals?
2. What is the antiderivative of \( \sec(x) \)?
3. Can you explain how the derivative of \( \sec(x) \) leads to \( \sec(x) \tan(x) \)?
4. How do we handle the absolute value in logarithmic antiderivatives?
5. What is the geometric interpretation of the antiderivative of trigonometric functions?

**Tip:** Remember that the constant \( C \) in indefinite integrals combines all arbitrary constants from each part of the integral.

Learn how to evaluate the indefinite integral of csc(x) + cot(x) + 2sec(x)tan(x). This step-by-step guide covers trigonometric integrals, including the antiderivatives of csc(x), cot(x), and sec(x)tan(x). Suitable for advanced high school or early college calculus students.

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