Integral of Sine and Cosine Product

Evaluate the integral $\int_{0}^{\frac{\pi}{3}} \sin(x) \cos\left(\frac{\pi}{3} - x\right) \, dx$.

This problem involves the evaluation of a definite integral where the integrand is a product of sine and cosine functions. To solve this, we can utilize the trigonometric identities to simplify the integrand. Recognizing and applying identities is a crucial strategy in integration when dealing with trigonometric functions, as it often makes the integral easier or even possible to solve analytically.

One particularly useful identity in this case is the product-to-sum identity, which can transform products of sine and cosine into sums of simpler trigonometric functions. This transformation is not only elegant but highly practical, as it reduces the problem into evaluating basic trigonometric integrals. Understanding how to manipulate these identities and switch perspectives—as from product form to sum form—can be an insightful process for tackling more complex integrals.

Furthermore, this problem requires knowledge of fundamental integration techniques. Once the integrand is simplified, the integration limits must be applied properly. Evaluational skills are important for calculating definite integrals, which represent the signed area under a curve. Such skills are foundational in calculus and are extensively applicable in physics and engineering fields where these mathematical tools are essential for modeling and solving real-world problems.