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1. **Integrate**  
   \[
   \int \frac{x^3 \, dx}{x^4 + 3x^2 + 2}
   \]
   (3 Marks)

2. **Integrate**  
   \[
   \int \frac{x \, dx}{(x - a)^2 (x - b)}
   \]
   (4 Marks)

3. **Obtain a reduction formula for**  
   \[
   \int (x^2 + a^2)^n dx
   \]
   (3 Marks)

4. **Evaluate from first principle**  
   \[
   \frac{d}{dx} \left( e^x \right)
   \]
   (3 Marks)

5. **Evaluate**  
   \[
   \lim_{n \to \infty} \left( \left(1 + \frac{1}{n}\right) \left(1 + \frac{2}{n}\right) \cdots \left(1 + \frac{n}{n}\right) \right)^{\frac{1}{n}}
   \]
   (3 Marks)

6. **Evaluate, if possible**  
   \[
   \int_{-\infty}^{\infty} \frac{dx}{1 + x^2}
   \]
   (4 Marks)

Would you like a step-by-step solution for a particular question, or should I solve them one by one? Let me know!

Here are some related questions to help further your understanding:

1. How would you approach integration by parts for complex integrals?
2. What techniques are useful when handling improper integrals?
3. What is the concept behind a reduction formula in integration?
4. How do you interpret limits involving infinite products?
5. Why is the derivative of \( e^x \) the same as the function itself?

**Tip:** For integrals, always consider whether substitution, partial fractions, or other standard methods apply before moving to more advanced techniques.

Integrate and evaluate several complex expressions involving polynomials and trigonometric functions.

This page provides solutions for advanced integration problems, including reduction formulas, limits, and improper integrals. Topics covered include polynomial integrals, the first principle of derivatives, and limits involving infinite products. Suitable for undergraduate math students.

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