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I'll help you fill out the table by identifying the best integration techniques for each integral and outlining any special steps involved.

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1. **Integral: \(\int \frac{\arctan x}{x^2 + 1} \, dx\)**  
   - **Technique**: **U-Substitution**  
   - **Special Steps**:  
     Let \( u = \arctan x \), then \( du = \frac{1}{x^2 + 1} \, dx \).

2. **Integral: \(\int (x + 2) \sqrt{x} \, dx\)**  
   - **Technique**: **Direct Integration** (Expand and integrate term by term)  
   - **Special Steps**:  
     Rewrite as \( \int (x^{1/2} + 2x^{1/2}) \, dx \), then integrate each term separately.

3. **Integral: \(\int \frac{e^x}{4 + e^{2x}} \, dx\)**  
   - **Technique**: **U-Substitution**  
   - **Special Steps**:  
     Let \( u = e^x \), then \( du = e^x \, dx \). This simplifies the denominator.

4. **Integral: \(\int \frac{1}{x \ln x} \, dx\)**  
   - **Technique**: **U-Substitution**  
   - **Special Steps**:  
     Let \( u = \ln x \), then \( du = \frac{1}{x} \, dx \).

5. **Integral: \(\int x^2 \cos(5x) \, dx\)**  
   - **Technique**: **Integration by Parts**  
   - **Special Steps**:  
     Use the formula \( \int u \, dv = uv - \int v \, du \). Let \( u = x^2 \) and \( dv = \cos(5x) \, dx \).

6. **Integral: \(\int \frac{x^2 + 1}{x} \, dx\)**  
   - **Technique**: **Direct Integration**  
   - **Special Steps**:  
     Simplify to \( \int (x + \frac{1}{x}) \, dx \), then integrate term by term.

7. **Integral: \(\int \frac{x + 5}{x^2 + 4} \, dx\)**  
   - **Technique**: **U-Substitution**  
   - **Special Steps**:  
     Let \( u = x^2 + 4 \), then \( du = 2x \, dx \). Handle \( \frac{x}{x^2 + 4} \) separately.

8. **Integral: \(\int \tan^4 x \sec^2 x \, dx\)**  
   - **Technique**: **U-Substitution**  
   - **Special Steps**:  
     Let \( u = \tan x \), then \( du = \sec^2 x \, dx \).

9. **Integral: \(\int x e^{5 - 2x} \, dx\)**  
   - **Technique**: **Integration by Parts**  
   - **Special Steps**:  
     Let \( u = x \) and \( dv = e^{5 - 2x} \, dx \), then apply integration by parts.

10. **Integral: \(\int \frac{1}{x^3 \sqrt{x^2 - 1}} \, dx\)**  
    - **Technique**: **Trig Substitution**  
    - **Special Steps**:  
      Use \( x = \sec \theta \), then \( dx = \sec \theta \tan \theta \, d\theta \), and simplify the square root term.

11. **Integral: \(\int \frac{x^5 + x - 1}{x^3 + 1} \, dx\)**  
    - **Technique**: **Partial Fractions**  
    - **Special Steps**:  
      Perform polynomial division first and then use partial fractions on the result.

12. **Integral: \(\int \frac{x^2 \sqrt{3x + 2x - x^2}}{x^2 - x} \, dx\)**  
    - **Technique**: **Simplification and U-Substitution**  
    - **Special Steps**:  
      Simplify the expression inside the square root and denominator, then apply a U-substitution or trig substitution based on simplification.

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Would you like any detailed steps for solving these integrals, or further clarifications?

### Related Questions:
1. How can we perform integration by parts for trigonometric functions like \(\int x^2 \cos(5x) \, dx\)?
2. When should we use trig substitution in integrals involving square roots?
3. Can you explain partial fraction decomposition in rational integrals?
4. How do we handle integrals with logarithmic functions such as \(\int \frac{1}{x \ln x} \, dx\)?
5. What are common U-substitution tricks for exponential functions like \(e^x\)?

### Tip:
In integration, recognizing patterns such as trigonometric identities or logarithmic differentiation helps streamline the problem-solving process.

For each integral, determine the best technique of integration to use (U-Sub, Parts, Trig Substitution, Partial Fractions, or direct integration). In the last column, indicate any special steps you would need to take in order to solve the integral (What u-substitution would you use? Which trig sub? Which trig identities?).

Learn how to determine the best integration technique for each integral. Includes strategies like U-Substitution, Integration by Parts, Trig Substitution, and Partial Fractions. Explore step-by-step methods and special steps to solve various integrals.

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