Evaluate Integral Using Trigonometric Substitution

Evaluate $\displaystyle \int_{0}^{\frac{1}{2}} \sqrt{1-x^2} \, dx$ using trigonometric substitution, where $x=\sin\theta$.

This problem involves the application of trigonometric substitution, a technique often employed to evaluate integrals containing radical expressions. The integral in question, involving the square root of a difference of squares, is a classical scenario for this method. By substituting x with sin of theta, we are able to transform the integral into a more manageable form. This substitution leverages the Pythagorean identity, which links trigonometric functions with algebraic expressions. The resulting integral, expressed in terms of theta, is typically easier to solve due to the intrinsic properties of trigonometric functions.

The strategy for employing trigonometric substitution relies on recognizing the integral's resemblance to a known trigonometric identity. For the given integrand, where we see a square root involving a sum or difference of squares, we can select a substitution that simplifies the expression by converting it into a trigonometric integral. The challenge lies in correctly choosing the substitution and performing the subsequent algebraic manipulations, such as applying derivative relationships and adjusting the limits of integration. This problem reinforces the importance of mastering trigonometric identities and honing algebraic manipulation skills, which are crucial in various fields of calculus and analysis.

In terms of approach, it's also critical to understand how the boundaries of integration transform under substitution. The limits, initially given in terms of x, need adjustment to reflect their values in terms of theta. Such transformations form a vital part of solving definite integrals using trigonometric substitutions and offer a perspective into the geometric interpretations of integrals.

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