Integral of x over square root of 25 minus x squared using Trigonometric Substitution

Evaluate the integral $\displaystyle \int \frac{x}{\sqrt{25 - x^2}} \, dx$ using trigonometric substitution.

In this problem, we are tasked with evaluating an integral that incorporates a square root with a quadratic expression in the form of 25 minus x squared. This structure is ripe for trigonometric substitution, a method that leverages the trigonometric identities and their derivatives to simplify the integration process. For integrals with expressions like the square root of a squared minus x squared, the common substitution uses the sine function due to its property of mapping the interval [-1, 1] to a complete period. This transforms the integral into one involving trigonometric functions, which are often easier to integrate due to their well-known derivatives and integrals.

The general strategy involves substituting x with a trigonometric expression such as 5 sin(theta), where 5 arises from the square root of 25 in the original expression. This substitution not only simplifies the square root but changes the limits of integration, effectively transforming the problem from one of algebraic form to a trigonometric form. This is beneficial because it introduces trigonometric identities that can further simplify the result. Once the integral in terms of theta is evaluated, we substitute back in terms of x to get the final result. This problem highlights the ability of trigonometric substitution to transform and simplify complex algebraic integrals into more manageable trigonometric integrals, underscoring its utility in calculus when faced with radicals of quadratic expressions.

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