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.