Indefinite Integral with Trigonometric Substitution2

Solve the indefinite integral $\displaystyle \int 4x^2 \sqrt{16 - x^2} \, dx$ using appropriate substitution.

This problem involves solving an indefinite integral by employing a technique known as trigonometric substitution. Trigonometric substitution is particularly useful when dealing with integrals that involve square roots of quadratic expressions. These expressions often resemble the identity properties of trigonometric functions, allowing us to simplify the integration process by converting the variables into trigonometric ones.

In this specific problem, the integral $\int 4x^2 \sqrt{16 - x^2} \, dx$ suggests a substitution that can transform the square root expression into a trigonometric identity. The function under the square root, $16 - x^2$, hints at a Pythagorean identity, typically involving sine or cosine. Consider the substitution $x = 4\sin(\theta)$, which then transforms the radical into a basic trigonometric function, simplifying the integration.

The overall strategy is to select a substitution that simplifies the integral into a more manageable form. The choice of substitution reflects an understanding of the mathematical identities between algebraic and trigonometric functions. This approach not only aids in the immediate problem but also enhances the problem-solver's ability to tackle a wide variety of integral problems where similar patterns occur. Through these exercises, students build fluency in recognizing these patterns and applying appropriate techniques efficiently.

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