Key Concept: Properties of Definite Integrals with Symmetric Limits

This problem leverages a crucial property of definite integrals, particularly useful for integrals with symmetric limits, i.e., from $-a$ to $a$. The property states: If $I = \int_{-a}^{a} f(x) \, dx$, then we can also write $I = \int_{-a}^{a} f(-x) \, dx$. Adding these two forms gives $2I = \int_{-a}^{a} [f(x) + f(-x)] \, dx$. This property is often referred to as a variation of King's Rule. It's especially powerful when $f(x) + f(-x)$ simplifies significantly.

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