To evaluate the integral $\int_{-1}^{1} \frac{\sin(x)}{x} \, dx$, we need to be aware that this integral is a well-known improper integral called the sine integral, denoted as $\text{Si}(x)$.

However, directly evaluating it from $-1$ to $1$ involves understanding the properties of the sine function and using symmetry.

Given the symmetry of $\sin(x)$ around $0$, the integral can be split as:

$\int_{-1}^{1} \frac{\sin(x)}{x} \, dx = 2 \int_{0}^{1} \frac{\sin(x)}{x} \, dx$

This is because $\frac{\sin(x)}{x}$ is an even function, so the integral from $-1$ to $1$ is twice the integral from $0$ to $1$.

To solve $\int_{0}^{1} \frac{\sin(x)}{x} \, dx$, we recognize that this is the definition of the Sine Integral function, $\text{Si}(x)$:

$\text{Si}(x) = \int_{0}^{x} \frac{\sin(t)}{t} \, dt$

Thus,

$\int_{0}^{1} \frac{\sin(x)}{x} \, dx = \text{Si}(1)$

The value of $\text{Si}(1)$ is a known constant, approximately:

$\text{Si}(1) \approx 0.946083$

Therefore,

$\int_{-1}^{1} \frac{\sin(x)}{x} \, dx = 2 \cdot \text{Si}(1) \approx 2 \cdot 0.946083 = 1.892166$

Thus, the integral evaluates to approximately 1.892166.

Would you like more details or have any questions?

Here are five questions you can ask next:

1. How do you integrate functions using integration by parts?
2. What is the process for solving definite integrals?
3. How do you handle improper integrals?
4. Can you explain the Sine Integral function in more detail?
5. What are some common techniques for evaluating difficult integrals?

Tip: When dealing with symmetric integrals, consider whether the function is even or odd to potentially simplify the computation.