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The problem asks which integrals are equal to zero without actually evaluating the integrals. Let's analyze each one based on properties of definite integrals, symmetries, and odd/even functions.

1. **\(\int_{-\pi/2}^{\pi/2} \sin(3x) \, dx\)**:
   - The sine function is an odd function, meaning \(\sin(-x) = -\sin(x)\).
   - The limits of integration are symmetric around zero, and the integrand is an odd function.
   - The integral of an odd function over a symmetric interval is zero.
   - **Conclusion: This integral is equal to zero.**

2. **\(\int_{-\pi/2}^{\pi/2} \left(\frac{1}{x} + \frac{1}{x^2}\right) dx\)**:
   - \(\frac{1}{x}\) is an odd function, while \(\frac{1}{x^2}\) is an even function.
   - The integral of an odd function over a symmetric interval is zero, but the integral of an even function is generally non-zero.
   - Therefore, this integral is not zero.
   - **Conclusion: This integral is not zero.**

3. **\(\int_{0}^{2\pi} \cos x \, dx\)**:
   - \(\cos(x)\) is an even function, but over one period from \(0\) to \(2\pi\), the integral of \(\cos(x)\) is zero because the positive and negative areas cancel out.
   - **Conclusion: This integral is equal to zero.**

4. **\(\int_{-\pi/4}^{\pi/4} x^3 e^x \, dx\)**:
   - \(x^3 e^x\) is an odd function because \(x^3\) is odd and \(e^x\) is even, so their product is odd.
   - Since the function is odd and the limits are symmetric around zero, the integral is zero.
   - **Conclusion: This integral is equal to zero.**

5. **\(\int_{-\infty}^{\infty} \left( \sin(5x) - \frac{1}{x} \sin x \right) dx\)**:
   - \(\sin(5x)\) is an odd function, and \(\frac{1}{x} \sin(x)\) is also an odd function (as both \(\frac{1}{x}\) and \(\sin(x)\) are odd).
   - The integral of an odd function over a symmetric interval is zero, but here the interval is from \(-\infty\) to \(\infty\), which requires more careful consideration.
   - For improper integrals involving functions like \(\frac{1}{x} \sin x\), the integral may not converge, so this integral does not necessarily equal zero.
   - **Conclusion: This integral is likely not zero.**

### Summary:
The integrals that are equal to zero are **1, 3, and 4**. This corresponds to option **B (1, 2)**.

Would you like more details on any specific step, or have any questions? Here are five questions to extend the discussion:

1. How do you determine if a function is odd or even?
2. What is the significance of the limits of integration being symmetric around zero?
3. Can an integral of an even function over a symmetric interval ever be zero?
4. How do you approach evaluating improper integrals like in question 5?
5. What are the conditions for an integral involving a product of an odd and even function?

**Tip:** Always consider the symmetry of the integrand when dealing with definite integrals, especially when the limits of integration are symmetric around zero.

Which of the following integrals are equal to zero? Note that you do not need to evaluate the integrals explicitly.

Learn how to identify which integrals are equal to zero without explicit evaluation. This problem explores definite integrals, the properties of odd and even functions, and symmetry. Suitable for undergraduate students studying calculus.

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