The expression you've provided involves evaluating the definite integral of the square of a function \( g(t) \) over the interval \([-1, 1]\) and then multiplying the result by \( \frac{1}{2} \).

The expression is:

\[
\frac{1}{2} \int_{-1}^{1} g^2(t) \, dt
\]

To evaluate this, you would need to know the specific form of the function \( g(t) \). However, I can outline the general steps to solve it:

### General Steps:
1. **Square the Function**: Calculate \( g^2(t) \) based on the specific form of \( g(t) \).
2. **Set Up the Integral**: Write out the definite integral with the limits \(-1\) to \(1\).
3. **Integrate**: Compute the integral \( \int_{-1}^{1} g^2(t) \, dt \).
4. **Multiply by \( \frac{1}{2} \)**: Finally, multiply the result of the integral by \( \frac{1}{2} \).

Without the specific function \( g(t) \), I cannot perform the exact integration. If you provide \( g(t) \), I can solve it in detail.

Would you like to proceed with a specific function for \( g(t) \), or do you have any other questions?

Here are some related questions you might consider:
1. What is the significance of squaring the function inside the integral?
2. How does the choice of \( g(t) \) affect the value of the integral?
3. What are some common techniques for evaluating definite integrals?
4. How do the limits of integration influence the outcome of the integral?
5. How would the result change if the integral were from \([-2, 2]\) instead of \([-1, 1]\)?

**Tip:** When dealing with definite integrals, it's helpful to understand the symmetry of the function, as it can simplify the calculation.

Understand how to evaluate the definite integral ∫[-1,1] 1/2 * g^2(t) dt with our step-by-step guide. Learn about squaring a function inside an integral and the impact of interval limits on integral results. Ideal for advanced high school students and anyone interested in calculus.

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