$$ 1 \leftarrow \quad $$ Use symmetry to evaluate the following integral.

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We can often use symmetry when the integration limits are symmetric (for example, from $$-a$$ to $$a$$). In particular, suppose we need to evaluate an integral of the form

$$
I = \int_{-a}^{a} f(x) \, dx.
$$

The key is to determine whether the function $$f(x)$$ is even or odd.

1. **Even functions:**  
A function is even if

$$
f(-x) = f(x) \quad \text{for all } x.
$$

If $$f(x)$$ is even, then

$$
I = \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx.
$$

2. **Odd functions:**  
A function is odd if

$$
f(-x) = -f(x) \quad \text{for all } x.
$$

If $$f(x)$$ is odd, then

$$
I = \int_{-a}^{a} f(x) \, dx = 0.
$$

Thus, to evaluate the given integral using symmetry, we should:

- Check the parity of $$f(x)$$.
- If $$f$$ is even, compute $$2 \int_{0}^{a} f(x) \, dx$$.
- If $$f$$ is odd, the integral is $$0$$.

If you provide the specific function $$f(x)$$ or the full integral statement, we can apply these symmetry properties to obtain the numerical result.
To evaluate the integral using symmetry, check if the function $$f(x)$$ is even or odd. If $$f(x)$$ is even, the integral from $$-a$$ to $$a$$ is twice the integral from $$0$$ to $$a$$. If $$f(x)$$ is odd, the integral is zero.

\( 1 \leftarrow \quad \) Use symmetry to evaluate the following integral.

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